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nonssoK or lUTBiMAncB and ibtboitomt or wasbihotos usmiaiTT, baxht louu. 

VOL. I. 





Enterwl, •cconling to Act of Ci^ngnm, in the year 1863, by 


Tn the Clerk'ii OAre of the Dintrlrt Court ttt the United SUtee for the EMtem 

DiHtrict uf PeoDMylvaoU. 


The methods of investigation adopted in this work are 
in accordance with what may be called the modem school 
of practical astronomy, or more distinctively the Ger- 
man school, at the head of which stands the unrivalled 
Bessel. In this school, the investigations both of the 
general problems of Spherical Astronomy and of the Theory 
of Astronomical Instruments are distinguished by the gene- 
rality of their form and their mathematical rigor. When 
approximative methods are employed for convenience in 
practice, their degree of accuracy is carefully determined by 
means of exact formulaB previously investigated ; the latter 
being developed in converging series, and only such terms 
of these series being neglected as can be shown to be insen- 
sible in the cases to which the formulaB are to be applied. 
And it is an essential condition of all the methods of com- 
putation from data furnished by observation, that the errors 
of the computation shall always be practically insensible in 
relation to the errors of observation : so that our results 
shall be purely the legitimate deductions from the observa- 
tions, and free from all avoidable error. 

It is another characteristic feature of modem spherical 
astronomy, that the final formulae furnished to the practical 
computer are so presented as seldom to require accompany- 
ing verbal precepts to distinguish the species of the unknown 
angles and arcs ; and this results, in a great measure, from 
the consideration of the general spherical triangle^ or that in 
which the six parts of the triangle are not subjected to the 



condition that they shall each be less than 180°, but may 
have any values less than 360°, all ambiguity as to their 
species being removed by determining them, when necessary, 
by two of their trigonometric functions, usually the sine and 
the cosine. This feature is mainly due to Gauss, and was 
prominently exhibited in his Theoria Motus Corporum Ckx- 
lestiuniy published in 1809. The English and American 
astronomers have been slow to adopt this manifest improve- 
ment ; in evidence of which I may remark that the general 
spherical triangle was not treated of in any work in the 
English language, so far a^ I know, prior to the publication 
of my Treatise on Plane and Spherical Trigonometry^ in the 
year 1850. In the present work, I assume the reader to be 
acquainted with this form of spherical trigonometry, and to 
accept its fundamental equations in their utmost generality. 

A third and eminently characteristic feature of modem 
astronomy, is the use which it makes, in all its departments, 
of the method of least squares, namely, that method of 
combining observations which shall give the moat pnjhaUe 
results, or which shall be exposed to the least probable errors. 
This method is also due to Gauss, who (though anticipated 
in the publication of one of its practical rules by Legendre) 
was the first to give a philosophical exposition of its princi- 
ples. The direct effect of this improvement is not only that 
the most probable result in each case is obtained, but also 
that the relative degree of accuracy of that result is deter- 
mined, and thus the degixje of confidence with which it may 
be received imd the weight which it may be allowed to have 
in subsequent discussions. Judiciously employed, it ser\'es 
to indicate when a particular process has reached the limit 
of accuracy which it can afford, thereby saves fruitless 
labor, directs inquiry into new channels, and contributes 
greatly to accelerate the progress of the science. 

Whilst the scnence has been rapidly advancing in Europe, 
wo have in this country not been idle. Two of the most 
important improvements in practical astronomy have had 


their origin in the United States,— the method of finding 
differences of geographical longitude by the electric telegraph, 
and that of finding the geographical latitude by the zenith 
telescope. These are the direct offspring of our admirably 
conducted Coast Survey, which, with the aid of these 
methods, both of the greatest simplicity, has fixed the lati- 
tudes and relative longitudes of a series of points on our 
coast with a degree of accuracy wholly unapproached in any 
previous work of this kind. This extreme accuracy will be 
apparent to the reader who will refer to the examples here 
given, which have been selected (almost at random) from 
the records of the Survey. 

It is perhaps necessary to say a few words here respect- 
ing those portions of this treatise in which I have ventured 
to substitute my own methods for those heretofore employed. 
My method of reducing lunar distances, which was first 
published in the American Ephemeris for 1855, is here re- 
produced, together with the necessary tables for its applica- 
tion. But I have first, for the sake of completeness, given 
the usual rigorous solution, although this is confessedly too 
laborious for ordinary use, and especially for use at sea. The 
approximative methods heretofore proposed may be divided 
into two classes : first, those based upon sufficiently precise 
formulsB, but such that the tables required in their applica- 
tion are adapted only to a mean state of the atmosphere; 
and second, those based upon incomplete formulsa. As to 
the first class, the trouble of correcting the tabular numbers 
for the barometer and thermometer would render the 
methods as laborious as the rigorous method, and it is 
therefore the usual practice, at sea, to disregard these correc- 
tions altogether, thus introducing a greater error than would 
follow from the use of the more incomplete formulaa of the 
second class, if in the latter these corrections were taken 
into account. But, as to the methods of the second class (of 
which there are several in common use), it will be found 
upon examination that the omitted terms of the formulaa 

are not so small as to be insensible even in relation to the 
rather large errors of observation which are unavoidable ip 
the use of the sextant. The defects of both classes are 
supposed to be avoided in my new method ; for, first, 1 have 
deduced a rigorous formula from which is derived an ap- 
proximate one, practically perfect, representmg the true cor- 
rection of the lunar distJinee within one second of arc in 
every cose that can occur in practice; and, second, I have 
arranged this formula so that it not only requires extremely 
simple tables in its application, but also tfie Utfmlar mtiiiJitrH 
retjuire iio correction for the Luromcter and thermometer, the 
corrections for the state of these instruments being intro- 
duced in a simple manner in fonning the anjumentu of the 
tables. In applying this method with It^arithms of only 
four decimal places, the true distance is usually obtained 
within less than two seconds of arc, a degree of accuracy fiu" 
greater than is necessary in relation to our present means 
of observing the distance. It is, in fact, quite as accurate 
m practice as Bessel's theoretically exact method when tlie 
latter is also carried out with four<place lugarithiits. I 
think, therefore, tinit I may justly prefer my own method 
not only to the imperfect approximative methods atwve 
referred to, but also tt) Bessel's method, which n-quires an 
extended Ephemeris wholly different from that now in use, 
and iti withal more lalmrious. 

The Gaussian method of reducuig circummcridian alti- 
tudi'H of the sun by n-ferring them lo the instant of the 
auw's maximum altitude, is in thi.s work rigorously investi- 
gated, and a small t*'nn, overlooked or disregarded by Gauss, 
]iaf( been added to the formula. 

A new and brief approximative method of finding the 
latitude by two altitudes near tlio meridian wlien the 
time is not known, is given in Vol. 1. Arts. 105 and 204, and 
another by thn« altitudes near tlie meridian, in Art- 205, 
which will probably 1k^ found useful as nautiejd nietb()ds. 

The subject of Eclipses will be fomid treated with more 


than usual completeness. The fundamental formulaB adopted 
are those of Bessel's theory, but the solutions of the various 
problems relating to the prediction of solar eclipses for the 
earth generally are mostly new. The rigorous solutions of 
these problems given by Bessel in his Analyse der Finster- 
nesse are not required for the usual purposes of prediction, 
however interesting they may be as specimens of refined 
and elegant analysis. On the other hand, the approximate 
solutions commoDly given appear to be unnecessarily rude. 
Those that I have substituted will be found to be very little 
if at all more laborious than the latter, while they are almost 
as precise as the former, and by a very little additional labor 
(that Is, by repeating only some parts of the computation 
for a second or third approximation) may be rendered quite 

So far as I can find, no one has heretofore treated distinct- 
ively of the occultations of planets by the moon, and these 
phenomena have been dismissed as simple cases of the 
general theory of eclipses, in which both the occulting and 
the occulted body are spherical. But in almost every oc- 
cultation of one of the principal planets, the planet will be 
either a spheroidal body fully or partially illuminated by 
the sun, or a spherical body partially illuminated : so that, 
in the general case, we have to consider the disc of the oc- 
culted body as bounded by an ellipse or by two different 
semi-eUipses. I have discussed this general case at length, 
and have adapted the theory to each planet specially. The 
additional computations required to take into account the 
true figure of the planet's disc are sufficiently brief and 
simple. The case of the occultation of a cusp of Venus or 
Mercury is included in the discussion, and also the occulta- 
tion of Saturn's rings. 

The well known formula for predicting the transits of the 
inferior planets over the sun's disc, first given by Lagrange, 
is here rendered more accurate by introducing a considera- 

tion of the compression of the earth ; and a uew and simple 
demonstration of the formula is given. 

In the practical portions of the work, and especially in 
the second volume, I have endeavored to give every import- 
ant precept for the guidance of observers, deduced from the 
labors of others or suggested by my own experience. All 
the principal methods are illustrated by examples from 
actual observation. 

I have taken especial pains throughout the work to ex- 
hibit the mode of discussing the probable errors of the results 
obtained by observations, and have given numerous examples 
of the application of tlie method of least squares. This 
method is applicable in almost all the physical sciences 
where numerical results are to ]»e deduced, and, therefore, 
does not necessarily form a part of a work on astronomy ; 
but, as I could not refer my reader to any work in the 
English language for a sufficient account of the method, I 
have prepared a concise treatise upon it, which forms the 
Appendix. In this, I have confined myself chiefly to the 
parts of the theory required in practical astronomy, and have 
endeavored to present its principles in a simple yet rigonius 
manner (so far as the subject allows), taking as a ba-sis 
known theorems of the calculus of probabilitiesj and follow- 
ing principally the processes first proposed by Gauss. 

In this Ap|>endix I have treated of Peirck's Criterion for 
the rejection of doubtful oljservatious, which is already well 
known to American aj^tronomers, and is now constantly 
applied in the discussion of observations upon our Coast 
Survey. Objectittns have Ijeen made to the criterion, l>ut 
none that would not apply equally well to the method of 
least stjuurcs itself. To tliu^* who have not been able to 
follow pErncE's investigation, the simple approximate cri- 
t4>rion which I have suggested at the end of the Ap{)endix 
may prove acceptable. It is derived directly from the fim- 
dauiental formula of the method of least stjuorcs, and leads 


to the rejection of nearly the same observations as that of 

The plates at the end of the work exhibit in minute 
detail the instruments now chiefly employed by astronomers. 
To have given more, with the necessary explanations, would 
have led me too far into the mere history of the subject, and 
would have occupied space which I thought it preferable to 
fill with discussions relating to the leading instruments now 
in use. The scale of these plates is purposely made quite 
small; but the great precision with which they are executed 
will enable the reader to measure from them the dimensions 
of all the important parts of each of the principal instru- 
ments. I am greatly indebted for the perfection of these 
drawings to the engravers, the Messrs. Illman Brothers, of 

Such auxiliary tables as seemed to be necessary to the 
reader in using these volumes have been given at the end 
of Vol. II. Some of these are new. Most of those which 
have been derived from other sources have been either re- 
computed or tested by differences and corrected. To insure 
their accuracy, they have also been tested by differences 
after being in type. 

For the very complete index to the whole work, I am 
indebted to my friend. Prof J. D. Crehore, of Washington 

In conclusion, I desire to express my obligations to those 
citizens of Saint Louis who, without solicitation, have gene- 
rously assumed a share of the risk of publication. Their 
liberal spirit has been met by a corresponding liberality on 
the part of my publishers, who have spared no expense in 
the typographical execution. I shall be content if their 
expectations are not wholly disappointed, and the work 
contributes in any degree to the advancement of the noblest 
of the physical sciences. 

Washington Unitebsitt, 

Saint Louis, January 1, 1863. 





The Celestial Sphbrb — Spherical and Rectangular Co-obdinates 17 

Spherical co-ordinates 18 

Transformation of spherical co-ordinates 27 

Rectangular co-ordinates. .. 43 

Transformation of rectangular co-ordinates 48 

Differential yariations of co-ordinates 50 


Time — Use of the Ephemeris — Interpolation — Star Catalogues 62 

Solar time 68 

Sidereal time 59 

Hour angles 64 

Ephemeris 68 

Interpolation by differences of any order 79 

Star catalogues 91 


Figure and Dimensions of the Earth 96 

Reduction of latitude 97 

Radius of the terrestrial spheroid for given latitudes 99 

Normal, &c 101 


Reduction of Obsbrtations to the Centre of the Earth 108 

Parallax 104 

Refraction. — General laws of refraction 127 

Tables of refraction 180 

Differential equation of the atmospheric refraction 186 

Integration of the differential equation with Bououer's hypothesis 186 

Integration with Bessel's hypothesis according to the methods of Krahp 

and Laplace 148 




Coiutruction of Bbssbl's Table 165 

Refraction in right ascension and declination 171 

Dip of the horiion 172 

Semidiameters of celestial bodies „ 180 

Augmentation of the moon's semidiameter 183 

Contraction of the sun's and the moon's semidiameters by refWu:tion 184 

Reduction of obserred zenith distances to the centre of the earth 189 


FiXDiNO THE Time by Asteonomical Observations 193 

1st Method.— Bj transits 1% 

2d Method. — Bj equal altitudes 190 

dd Method. — By a single altitude or zenith distance 200 

Correction for second differences of zenith distance 213 

4th Method. — By the disappearance of a star behind a terrestrial object.... 217 

Time of rising and setting of the stars 218 


Ist Method. — By a single altitude 219 

2d Method.— By equal altitudes 220 


FiXDiBO THE Latitude bt Astbonomical Obsbbvations 223 

1st Method. — By meridian altitudes or zenith distances....... 223 

Combination of pairs of stars whose meridian zenith distances are 

nearly equal (see Vol. II., Zenith Telescope) 226 

Meridian altitudes of a circumpolar star 226 

Meridian zenith distances of the sun near the solstices 228 

2d Method. — By a single altitude at a given time 229 

8d Method. — By reduction to the meridian when the time is given 238 

Circummeridian altitudes 235 

Gauss's method of reducing circummeridian altitudes of the sun 244 

Limits of the redaction to the meridian 261 

4ih Method.— By the Pole Star 263 

6th Method. — By two altitudes of the same star, or different stars, and the 

elapsed time between the observations.. 267 

General solution 268 

Caillet's formulsD for a fixed star or the sun 264 

Correetionof this method for the sun 266 

6th Method. — By two altitudes of the same or different stars, with the 

difference of their azimuths .« 277 

7Ui Method. — By two different stars observed at the same altitude, when 

the time is given 277 

At nearly the same altitude, observed with the zenith telescope 279 

8th Method. — By three stars observed at the same altitude (Gauss's method) 280 

The same by Caoxoli's formulsB 286 

By a number of stars observed at the same altitude, treated by the 

Method of Least Squares 280 

9th Method. — By the transits of stars over vertical circlet (see Vol. II., 

Transit Instrument in the Prime Vertieml) ~ 298 


10th Method. — By altitudes ne&r the meridian when the time is not known... 296 
(A.) By two altitudes near the meridian and the chronometer times of 
the obserTations, when the rate of the chronometer is known, but not 

its correction 296 

(B.) By three altitudes near the meridian and the chronometer times 
of the obseryations, when neither the correction nor the rate of the 

chronometer is known 299 

(C.) By two altitudes near the meridian and the difference of the 

azimuths 801 

(D.) By three altitudes near the meridian and the differences of 

azimuths 802 

11th Method. — By the rate of change of altitude near the prime vertical 808 

FiNDiMO THB Latitude at Ska 804 

1st Method. — By meridian altitudes 804 

2d Method. — By reduction to the meridian when the time is given 307 

8d Method. — By two altitudes near the meridian when the time is not 

known 807 

4th Method. — By three altitudes near the meridian when the time is not 

known 809 

6th Method. — By a single altitude at a given time... 810 

6th Method. — By the change of altitude near the prime vertical 811 

7th Method.— By the Pole Star 311 

8th Method. — By two altitudes with the elapsed time between them 813 


Finding the Longitude bt Astronomical Observations 817 

1st Method. — By portable chronometers 317 

Chronometrio expeditions 823 

2d Method.— By signals 887 

Terrestrial signals 387 

Celestial signals, (a) Bursting of a meteor. (6) Beginning or end- 
ing of an eclipse of the moon, (c) Eclipses of Jupiter's satellites. 
{d) Occultations of Jupiter's satellites, (e) Transits of the satel- 
lites over Jupiter's disc. (/) Transits of the shadows of the satel- 
lites over Jupiter's disc, (g) Eclipses of the sun, Occultations of 

stars and planets by the moon. [See Chapter X.] 389 

8d Method. — By the electric telegraph 341 

Method of star signals 842 

4th Method. — By moon culminations 850 

Pbiroe's method of correcting the epbemeris 358 

Combination of moon culminations by weights 368 

5th Method. — By azimuths of the moon, or transits of the moon and a star 

over the same vertical circle 871 

6th Method. — By altitudes of the moon 882 

(A.) — By the moon's absolute altitude 388 

(B.) — By equal altitudes of the moon and a star observed with the 

Zenith Telescope 386 

7th Method. — By lunar distances 893 

(A.) — Rigorous method 395 

(B.) — Approximative method 402 



Bj chronometers « 420 

By lunar distances ■ 422 

Bj the eclipses of Jupiter's satellites ^ 428 

Bj the moon*s altitude 428 

Bj occultations of stars by the moon 424 

FiXDixo A Ship's Place at Sea bt Circles of Positiox — Sumscer's Method... 424 

The Meridulh Lixe axd Variatiox or tub Compass 429 


Eclipses 486 

Solar Eclipses. Prediction for the earth generally 486 

Fundamental equations 489 

Outline of the shadow 456 

Rising and setting limits 466 

Cunre of maximum in horizon 475 

Northern and southern limits 480 

Cunre of central eclipse 491 

Limits of total or annular eclipse 498 

Prediction for a giTen place 505 

Correction for atmospheric refraction in eclipses 515 

Correction for the height of the obserrer aboTe the lerel of the sea 517 

Application of obserred solar eclipses to the determination of terrestrial longi- 
tudes and the correction of the elements of the computation 518 

Lunar eclipses 542 

Occultations of ftxed stars by the moon 549 

Terrestrial longitudes from occultations of stars... 550 

Prediction of occultations 557 

Limiting parallels.. 561 

Occultations of planets by the moon 565 

Apparent form of a planet's disc 566 

Terrestrial longitude from occultations of planets 578 

Transits of Venus and Mercury - 591 

Determination of the solar parallax 592 

Prediction for the earth generally •••• 598 

Occultation of a fixed star by a planet 601 



Stars 002 

Precession ^*>0A 

Nutation 624 

Aberration 628 

ParmlUz 643 

Mmii RBd appttrent places of stars 645 



Deterxixatioh of the Obliquity of the Ecliptic and the Absolute Right 

Ascensions and Declinations of Stars by Observation 658 

Obliqaitj of the ecliptic 069 

Equinoctial points, and absolute right ascension and declination of the fixed 
stars 666 


Determination of Astronomical Constants by Observation 671 

Constants of refiraotion 671 

Constant of solar parallax 678 

Constant of lunar parallax 680 

Mean semidiameters of the planets 687 

Constant of aberration and heliocentric parallax of fixed stars 688 

Constant of nutation 698 

Constant of precession 701 

Motion of the sun in. space 703 





1. From whatever point of space an observer be supposed to 
view the heavenly bodies, they will appear to him as if situated 
upon the surface of a sphere of which his eye is the centre. If, 
without changing his position, he directs his eye successively to 
the several bodies, he may learn their relative directions, but 
cannot determine either their distances from himself or from 
each other. 

The position of an observer on the surface of the earth is, 
however, constantly changing, in consequence, 1st, of the diur- 
nal motion, or the rotation of the earth on its axis ; 2d, of the 
annual motion, or the motion of the earth in its orbit around 
the sun. 

The changes produced by the diurnal motion, in the appa- 
rent relative positions or directions of the heavenly bodies, are 
different for observers on different parts of the earth's surface, 
and can be subjected to computation only by introducing the 
elements of the observer's position, such as his latitude and 

But the changes resulting from the annual motion of the 
earth, as well as from the proper motions of the celestial bodies 
themselves, may be separately considered, and the directions 
of all the known celestial bodies, as they would be seen from 
the centre of the earth at any given time, may be computed 

Vol. L— 2 17 


according to the laws which have been found to govern the 
motions of these bodies, from data furnished by long series of 
observations. The complete investigation of these changes and 
their laws belongs to Physical Astronomy^ and requires the consi- 
deration of the distances and magnitudes as well as of the direc- 
tions of the bodies composing the system. 

Spherical Astronomy treats specially of the directions of the 
heavenly bodies ; and in this branch, therefore, these bodies are 
at any given instant regarded as situated upon the surface of a 
sphere of an indefinite radius described about an assumed 
centre. It embraces, therefore, not only the problems which arise 
from the diurnal motion, but also such as arise from the annual 
motion so far as this aftects the apparent positions of the hea- 
venly bodies upon the celestial sphere, or their directions from 
the assumed centre. 


2. The direction of a point may be expressed by the angles 
which a line drawn to it from the centre of the sphere, or point 
of observation, makes with certain fixed lines of reference. But, 
since such angles are directly measured by arcs on the surface 
of the sphere, the simplest method is to assign the position in 
which the point appears when projected upon the surface of the 
sphere. For this purpose, a great circle of the sphere, supposed 
to be given in position, is assumed as a primitive circle of refer- 
ence, and all points of the surface are referred to this circle by a 
system of secondaries or great circles perpendicular to the primi- 
tive and, consequently, passing through its poles. The position 
of a point on the surface will then be expressed by two spherical 
co-ordinates: namely, 1st, the distance of the point from the pri- 
mitive circle, measured on a secondary ; 2d, the distance inter- 
cepted on the primitive between this secondary and some given 
point of the primitive assumed as the origin of co-ordinates. 

AVe shall have diflferent systems of co-ordinates, according to 
the circle adopted as a primitive circle and the point assumed as 
the origin. 

3. First system of co-ordinates. — Altitude aiid azimuth. — ^In this 
system, the primitive circle is the horizon^ which is that great 
circle of the sphere whose plane touches the surface of the 


earth at the observer.* The plane of tlie horizon may be con- 
ceived as that which sensibly coincides with the surface of a 
fluid at rest. 

The vertical line is a straight line perpendicular to the plane 
of the horizon at the observer. It coincides with the direction 
of the plumb line, or the simple pendulum at rest. The two 
points in which this line, infinitely produced, meets the sphere, 
are the zenith and nadir, the first above, the second below the 

The zenith and nadir are the poles of the horizon. 

Secondaries to the horizon are vertical circles. They all pass 
through the zenith and nadir, and their planes, which are called 
vertical planes, intersect in the vertical line. 

Small circles parallel to the horizon are called almucantarSj or 
parallels of altitude. 

The celestial meridian is that vertical circle whose plane passes 
through the axis of the earth and, consequently, coincides with 
the plane of the terrestrial meridian. The intersection of this 
plane with the plane of the horizon is the meridian line, and the 
points in which this line meets the sphere are the north and south 
points of the horizon, being respectively north and south of the 
plane of the equator. 

The prime vertical is the vertical circle which is perpendicular 
to the meridian. The line in which its plane intersects the 
plane of the horizon is the east and west line, and the points in 
which this line meets the sphere are the ea^t and west points of 
the horizon. 

The north and south points of the horizon are the poles of the 
prime vertical, and the east and west points are the poles of the 

* In this definition of the horixon we consider the plane tangent to the earth's 
surface as sensibly coinciding with a parallel plane passed through the centre ; that 
is, we consider the radius of the celestial sphere to be infinite, and the radius of the 
earth to be relatirely zero. In general, any number of parallel planes at finite dis- 
tances must be regarded as marking out upon the infinite sphere the same great circle. 
Indeed, since in the celestial sphere we consider only direction^ abstracted from dis- 
tance, all lines or planes haying the same direction — that is, all parallel lines or 
planes — must be regarded as intersecting the surface of the sphere in the same 
point or the same great circle. The point of the surface of the sphere in which a 
number of parallel lines are concciyed to meet is called the vanishing point of those 
lines ; and, in like manner, the great circle in which a number of parallel planes are 
conceiTed to meet may be called the vanishing circle of those planes. 


The altitude of a point of the celestial sphere is its distance 
from the horizon measured on a vertical circle, and its azimuth is 
the arc of the horizon intercepted between this vertical circle 
and any point of the horizon assumed as an origin. The origin 
from which azimuths are reckoned is arbitrary ; so also is the 
direction in which they are reckoned; but astronomers usually 
take the south point of the horizon as the origin, and reckon 
towards the right hand, from 0° to 360® ; that is, completely 
around the horizon in the direction expressed by writing the 
cardinal points of the horizon in the order S.W. N. P]. We 
may, therefore, also define azimuth as the angle which the 
vertical i)lane makes with the plane of the meridian. 

Navigators, however, usually reckon the azimuth from the 
uorth or south points, according as they are in north or south 
latitude, and towards the east or west, according as the point 
of the sphere considered is east or west of the meridian: so that 
the azimuth never exceeds 180®. Thus, an azimuth which is 
exjiressed according to the first method simply by 200^ would 
he expressed by a navigator in north latitude by X. 20® E., and 
l>y a navigator in south latitude by S. 160® E., the letter prefixed 
denoting the origin, and the letter aflSxed denoting the direction 
in which the aziuiuth is reckoned, or whether the point consi- 
dered is east or west of the meridian. 

When the point considered is in the horizon, it is often 
referred to the east or west points, and its distance from the 
nearest of these points is called its amplitude. Thus, a point in 
the horizon whose azimuth is 110® is said to have an amplitude 
of W. 20® X. 

Since by the diurnal motion the obsen'cr's horizon is made 
to change its position in the heavens, the co-ordinates, altitude 
and azimuth, are continually changing. Their values, therefore, 
will depend not only upon the observer's position on the earth, 
but upon the time reck(med at his meridian. 

In>tcad of the altitude of a point, we frequently employ its 
ze)tith diManee^ which is the arc of the vertical circle between the 
l»oint and the zenith. The altitude and zenith distance are, 
therefore, complements of each other. 

Wc shall hereafter denote altitude by h, zenith distance by ^, 
azimuth bv -4. AVe shall have then 

C =^ 90^ — /i /i = 90® — C 


The value of ^ for a point below the horizon will be greater 
than 90^, and the corresponding value of /i, found by the for- 
mula h = 90° — ^, will be negative : so that a negative altitude 
will express the depression of a point below the horizon. Thus, 
a depression of 10° \\\\\ be expressed by h = —10°, or ^ = 100°. 

4. Senovd system of co-ordinates. — Declination and hour angle, — In 
this system, the primitive circle is the celestial equator^ or that 
great circle of the sphere whose plane is perpendicular to the 
axis of the earth and, consequently, coincides with the plane of 
the terrestrial equator. This circle is also sometimes called the 

The diurnal motion of the earth does not change the position 
of the plane of the equator. The axis of the earth produced to 
the celestial sphere is called the axis of the heavens: the points 
in which it meets the sphere are the north and south poles of 
the equator, or the poles of the heavens. 

Secondaries to the equator are called circles of declination, and 
also hour circles. Since the plane of the celestial meridian 
passes through the axis of the equator, it is also a secondarj^ to 
the equator, and therefore also a circle of declination. 

Parallels of declination are small circles parallel to the equator. 

The declination of a point of the sphere is its distance from the 
equator measured on a circle of declination, and its hour angle is 
the angle at either pole between this circle of declination and the 
meridian. The hour angle is measured hy the arc of the equator 
intercepted between the circle of declination and the meridian. 
As the meridian and equator intersect in two points, it is neces- 
sary to distinguish which of these points is taken as the origin 
of hour angles, and also to know in what direction the arc which 
measures the hour angle is reckoned. Astronomers reckon 
from that point of the equator which is on the meridian above 
the horizon, towards the west, — that is, in the direction of the 
apparent diurnal motion of the celestial s])here, — and from 0° to 
360°, or from 0* to 24*, allowing 15° to each hour. 

Of these co-ordinates, the declination is not changed by the 
diurnal motion, while the hour angle depends only on the time 
at the meridian of the observer, or (which is the same thing) on 
the position of his meridian in the celestial sphere. All the 
observers on the same meridian at the same instant will, for the 
same st^r, reckon the same declination and hour angle. AVe have 


thus introduced coordinates of which one is wholly independent 
of the observer's position and the other is independent of his 

AVe shall denote declination by 5, and north declination will 
be distinguished by prefixing to its numerical value the sign +> 
and south declination by the sign — . 

We shall sometimes make use of the polar distance of a point, 
or its distance from one of the poles of the equator. If we denote 
it by P, the north polar distance will be found by the formula 

and the south polar distance by the formula 

The hour angle will generally be denoted by i. It is to be 
observed that as the hour angle of a celestial body is continually 
increasing in consequence of the diurnal motion, it may be con- 
ceived as having values greater than 360°, or 24*, or greater than 
any given multiple of 360°. Such an hour angle may be re- 
gjirdod as expressing the time elai)sed since some given passage 
of the body over the meridian. But it is usual, when values 
greater than 360° result from any caleuhition, to deduct 360°. 
Again, since hour angles reckoned towards the west are always 
positive, hour-angles reckoned towards the east must have the 
negative sign : so that an hour angle of 300°, or 20*, may also be 
expressed by — 60°, or 

r>. Third system of co-ordinates. — Declination and right ascoision. — 
In this system, the primitive phme is still the equator, and the 
first co-ordinate is the same as in the second svstem, namelv, the 
declination. The second co-ordinate is also measured on the 
e([uator, but from an origin which is not affected by the diurnal 
motion. Any i)oint of the celestial ecjuator might be assumed 
as the origin; but that which is most natunilly indicated is 
the vernal equinox, to define which some preliminaries are 

The ecliptic is the great circle of the celestial sjihere in which 
the sun ajjpears to move in consequence of the earth's motion in 
its orbit. The position of the ecliptic is not absolutely fixed in 
space; but, according to the definition just given, its i>osition at 
any instant coincides with that of the great circle in which the 


Buu appears to be moving at that instant. Its annual change is, 
however, very small, and its daily change altogether insensible. 

The obliquity of the ecliptic is the angle which it makes with 
the equator. 

The points where the ecliptic and equator intersect are called 
the equinoctial povits, or the equinoxes ; and that diameter of the 
sphere in which their planes intersect is the line of equinoxes. 

The vernal equinox is the point through which the sun ascends 
from the southern to the northern side of the equator ; and the 
autumnul equinox is that through which the sun descends from the 
northern to the southern side of the equator. 

The solstitial points^ or solstices^ are the points of the ecliptic 
90° from the equinoxes. They are distinguished as the north- 
ern and southern, or the summer and winter solstices. 

Tlie equinoctial colure is the circle of declination which passes 
through the equinoxes. The solstitial colure is the circle of decli- 
nation which passes through the solstices. The equinoxes are 
the poles of the solstitial colure. 

By the annual motion of the earth, its axis is carried very 
nearly parallel to itself, so that the plane of the equator, which 
is always at right angles to the axis, is very nearly a fixed plane 
of the celestial sphere. The axis is, however, subject to small 
changes of direction, the effect of which is to change the 
position of the intersection of the equator and the ecliptic, and 
hence, also, the position of the equinoxes. Li expressing the 
positions of stars, referred to the vernal equinox, at any given 
instant, the actual position of the equinox at the instant is 
understood, unless otherwise stated. 

The right ascension of a point of the sphere is the arc of the 
equator intercepted between its circle of declination and the 
vernal equinox, and is reckoned from the venial equinox east- 
ward from 0° to 360°, or, in time, from 0* to 24\ 

The point of observation being supposed at the centre of the 
earth, neither the declination nor the right ascension will be 
affected by the diurnal motion: so that these co-ordinates are 
wholly independent of the obsers'^er^s position on the surface of 
the earth. Their values, therefore, vary only with the time, 
and are given in the ephemeridcs as functions of the time 
reckoned at some assumed meridian. 

We shall generally denote right ascension by a. As its value 
reckoned towards the east is positive, a negative value resulting 


from any calculation would be interpreted as signifying an arc 
of the equator reckoned from the vernal equinox towards the 
west. Thus, a point whose right ascension is 300®, or 20*, may 
also be regarded as in right ascension — 60®, or — 4* ; but such 
negative values are generally avoided by adding 360®, or 24*. 
Again, in continuing to reckon eastward we may arrive at 
values of the right ascension greater than 24*, or greater than 
48*, etc.; but in such cases we have only to reject 24*, 48*, etc. 
to obtain values which exj^ress the same thing. 

6. Fourth system of co-ordinates, — Celestial latitude and langi- 
tude, — In this system the ecliptic is taken as the primitive circle, 
and the secondaries by which points of the sphere are referred 
to it are called circles of hititude. Parallels of latitude are small 
circles parallel to the ecliptic. 

The latitude of a point of the sphere is its distance from the 
ecliptic mea8ured on a circle of latitude, and its longitude is the 
arc of the ecliptic intercepted between this circle of latitude and 
the vernal equinox. The longitude is reckoned eastward from 
0® to 3G0®. The longitude is sometimes expressed in si(/ns^ 
degrees, &e., a sign being equal to 30®, or one-twelfth of the 

These co-ordinates are also independent of the diurnal motion. 
It is evident, however, that the system of declination and right 
ascension will be generally more convenient, since it is more 
directly related to our first and second systems, which involve 
the observer's position. 

AVe shall denote celestial latitude by /9; longitude by L Posi- 
tive values of ^3 belong to j)oints on the same side of the ecliptic 
as the north [jole; negative values, to those on the oi»posite side. 

In connection with this system we may here define the nana- 
gcsiintd, which is that point of the ecliptic which is at the greatest 
altitude above the horizon at any given time. That vertical 
circle of the observer which is jierpendicular to the ecliptic meets 
it in the nonagesimal : and, being a secondary to the ecliptic, it 
is also a circle of latitude: it is the great circle which passes 
through the observer's zenith and the pole of the ecliptic. 

7. Co-ordinates of the ohsnTers position. — We have next to ex- 
press the position of the obser\er on the surface of the earth, 
according to the different systems of co-ordinates. This will bo 


done by referring his zenith to the primitive circle in the same 
manner as in the case of any other point. 

In the first system, the primitive circle being the horizon, of 
which the zenith is the pole, the altitude of the zenith is always 
90°, and its azimuth is indeterminate. 

In the second system, the declination of the zenith is the same 
as the terrestrial latitude of the observer, and its hour angle is 
zero. The declination of the zenith of a place is called the 
geographical latitude^ or simply the latitude, and will be hereafter 
denoted by f. North latitudes will have the sign + ; south 
latitudes, the sign — . 

In the third system, the declination of the zenith is, as before, 
the latitude of the observer, and its right ascension is the same 
as the hour angle of the vernal equinox. 

In the fourth system, the celestial latitude of the zenith is the 
same as the zenith distance of the nonagesimal, and its celestial 
longitude is the longitude of the nonagesimal. 

It is evident, from the definitions which have been given, that 
the problem of determining the latitude of a place by astro- 
nomical observation is the same as that of determining the 
declination of the zenith ; and the problem of finding the lon- 
gitude may be resolved into that of determining the right 
ascension of the meridian at a time when that of the prime 
meridian is also given, since the longitude is the arc of the 
equator intercepted between the two meridians, and is, conse- 
quently, the diflerence of their right ascensions. 

8. The preceding definitions are exemplified in the following 

Fig. 1 is a stereographic projection of 
the sphere upon the plane of the horizon, 
the projecting point being the nadir. Since 
the planes of the equator and horizon are 
both perpendicular to that of the meridian, w\^ 
their intersection is also peri^endicular to 
it; and hence the equator WQE passes 
through the east and west points of the 
horizon. All vertical circles passing 
through the projecting point will be projected into straight 
lines, as the meridian NZS, the prime vertical WZE, and the 
vertical circle ZOH drawn through any point of the surface 


of the sphere. We have then, according to the notation adopted 
in the first system of co-ordinates, 

h ^ the altitude of the point ^= OS, 
C ^ the zenith distanco " ^^ OZ, 
A = the azimuth " = SM, or 

= the angle SZH. 
If tlie declination circle POD be drawn, we have, in the second 
aystcin of co-ordinates, 

3 ^ the declination of ^= OD, 
P = the polar distance " = PQ, 
I = the hour angle " = ZP2), or = QD. 
If Vis the vernal equinox, wo liavo, iii tlie third system of 

S = the declination of = OB, 
a = the riglit aeconBion = VD, or 
= the angle VPD. 
In this figure is also drawn the six hour circle EPW, or the 
declination circle passing through the east and west points of the 
horizon. The angle ZPW, or the arc QW, heing 90°, the hour 
angle of a point on this circle is either + 6* or —6*, that is, either 
6* or 18*. 
Fig. 2 is a repetition of the preceding figure, with the addi- 
ng. :. tion of the ecliptic and the circles related 
to it. C Vr re]) resents the ecliptic, P' its 
pole, P'OX a circle of latitude. Ilonce we 
have, in our third system of co-oi-dinates. 

^ = the celestial latitude of — OL, 
i ^^ " longitude " :^ VL, 

^ the angle VP'L. 

■\Ve have also FPthc eqninoetial colure,P'7'.4iJ the solstitial 
coliirc. P'ZGF the vcrticiil circle passing through i", which is 
therefore peri>ondicnlar to the ecliptic at G. The point G is the 
nonagesimal ; ZG is its zenith distance, Vtf its longitude; or 
ZG is the celestial latitude and VG the celestial longitude of the 
Finally, we have, in hoth Fig. 1 and Fig. 2, 

fi = the geograpliicnl latitude of the observer 
= ZQ -- 00" - PZ^ PX 


Hence the latitude of the observer is always equal to the alti- 
tude of the north pole. For an observer in south latitude, the 
north pole is below the horizon, and its altitude is a negative 
quantity: so that the definition of latitude as the altitude of the 
north pole is perfectly general if we give south latitudes the 
negative sign. The south latitude of an observer considered 
independently of its sign is equal to the altitude of the south 
pole above his horizon, the elevation of one pole being always 
equal to the depression of the other. 

9. Numerical expression of hour angles. — The equator, upon 
which hour angles are measured, may be conceived to be divided 
into 24 equal parts, each of which is the measure of one hour, 
and is equivalent to ^ of 360°, or to 15°. The hour is divided 
sexagesimally into minutes and seconds of time^ distinguished 
from minutes and seconds of arc by the letters "* and ' instead 
of the accents ' and ". We shall have, then, 

1* = 15° 1- = 15' 1- = 15" 

To convert an angle expressed in time into its equivalent in 
arc, multiply by 15 and change the denominations * "* * into 
° ' ''; and to convert arc into time, divide by 15 and change ° ' " 
into * »» •. The expert computer will readily find ways to 
abridge these operations in practice. It is well to observe, for 
this purpose, that from the above equalities we also have, 

lo = 4- 1' = 4- 

and that we may therefore convert degrees and minutes of arc 
into time by multiplying by 4 and changing ° ' into "^ * ; and 


10. Given the altitude (h) and azimuth {A) of a star, or of any point 
of the sphere, and the latitude (^) of the observer, to find the declina- 
tion (3) and hour angle {t) of the star or the point. In other words, 
to transform the co-ordinates of the first system into those of the 

This problem is solved by a direct application of the formulae 
of Spherical Trigonometry to the triangle POZ, Fig. 1, in which, 
being the given star or point, we have three parts given, 



Fig. 3. 

namely, ZO the zenith distance or complement of the given 
altitude, PZO the supplement of the given azimuth, and PZ the 
complement of the given latitude ; from which we can find PO 
the jv)lar distance or complement of the required declination, 
and ZPO the required hour angle. But, to avoid the trouble of 
taking complements and supplements, the fomiulfe are adapted 
8o as to ex[)ress the deelinati<m and hour angle directly in terms 
of the altitude, azimuth, and latitude. 

To show as clearly as possible how the formulje 
of Spherical Trisconometrv are thus converted into 
fonnuhe of Spherical Astronomy, let us first con- 
sider a spherical triangle ABC, Fig. 3, in which 
there are given the angle A, and the sides 6 and e, to 
find the angle B and the side a. The general rela- 
tions between these five quantities are [Sph. Trig. 
Art. 114]* 

cos a = cos c cos h -\- sin c sin b cos A ^ 
sin a cos B r^ sin c cos b — cos c sin b cos A > ((X) 
sin a sin B -~ sin b sin A J 

Now, comparing the triangle ABC with the triangle PZO of 
Fig. 1, we have 

A r^ PZO r= 180° — .4 
b=^ ZO^ 90°— h 
c= PZ= 90°— sp 

(3=^ PO:^90° — ^ 
B ^ PZO --^ t 

Substituting these values in the above equations, we obtain 

sin ^ - sin 9" sin h — cos ^ cos h cos A (1) 

cos d cos t r^ cos if H\n h -{- sin ^ cos /* cos A (2) 

cos d sin t ~- cos h sin A (3) 

which are the required expressions of o and t in terms of h and A, 
If the zenith distance (^) of the star is given, the equations 
will be 

sin o - : sin <p cos Z — cos if sin C cos A (4) 

cos a cos t —-. cos <f cos C + sin ^ sin r cos A (5) 

cos «J sin f 1 - sin C sin ^l (<>) 

Since, in Spherical Astronomy, we consider arcs and angles 
whose values mav exceed 180^, it becomes necessary, in general. 

♦ The references to Trigonometry are to the oth edition of the author's *'Trcnlisc 
on Plane and Spherical Trigonometry/' 


to determine such arcs and angles by both the sine and the 
cosine, in order to fix the quadrant in which their vahies are to 
be taken. It has been shown in Spherical Trigonometry that 
when we consider the general triangle, or that in which values 
are admitted greater than 180°, there are two solutions of the 
triangle in every case, but that the ambiguity is removed and 
one of these solutions excluded " when, in addition to the other 
data, the sign of the sine or cosine of one of the required parts is 
given." [Sph. Trig. Art. 113.] In our present problem the sign 
of cos 8 is given, since it is necessarily positive ; for d is always 
numerically less than 90°, that is, between the limits +90° and 
—90°. Hence eos t has the sign of the second member of (2) or 
(5), and sin t the sign of the second member of (3) or (6), and t 
is to be taken in the quadrant required by these two signs. Since 
h also falls between the limits +90° and —90°, or ^ between 0° 
and 180°, cos A, or sin ^, is positive, and therefore by (3) or (6) 
sin t has the sign of sin A ; that is, when A < 180° we have t < 
180°, and when A>180° we have ^ > 180°,— conditions which 
also follow directly from the nature of our problem, since the 
star is west or east of the meridian according as A < 180° or A 
> 180°. The formula (1) or (4) fully determines 5, which will 
always be taken less than 90°, positive or negative according to 
the sign of its sine.* 

To adapt the equations (4), (5), and (6) for logarithmic compu- 
tation, let m and M be assumed to satisfy the conditions [PL 
Trig. Art. 174], 

m sin Mz= sin C cos A 1 . 

m cos Jf = eos C / 

the three equations may then be written as follows : 

sin ^ = wi sin (<p — M) 
eos d COS t = m cos (^ — ilf) \ (8) 

cos ^ sin ^ = sin C sin A 

K we eliminate m from these equations, the solution takes the 
following convenient form : 

* There are, however, special problems in which it is convenient to depart from 
this general method, and to admit declinations greater than 90°, as will be seen 


tan Jlf = tan C cos A 

tan A sin M 

tan t = \ (^\ 

cos (jp — J[f ) ' ^^^ 

tan ^ = tan (tp — M) cos t 

in the use of which, we must observe to take t greater or less 
than 180° according as A is greater or less than 180°, since the 
hour angle and the azimuth must fall on the same side of the 

Example. — ^In the latitude f = 38° 58' 53", there are given for 
a certain star C =- 69° 42' 30", A = 300° 10' 30" ; required d and L 
The computation by (9) may be arranged as follows :* 

log tan C 0.4320966 

^ r^ 38® 68^ bS'^ log COB A 9.7012595 log tan A ii0.2866026 

M= 53 39 41.98 log tan i/^ 0.133356]_ log sin .If 9.9060828 

^— Jf= — 14 40 48.98 logtan (0— if) n9!4182633 log sec (^—M) 0.0144141 

/= 304 55 26.49 log cos ^ 9.7577677 log tan t fiO.1559995 

i5= —8 81 46.50 log tan (J n9.1760310 

Converting the hour angle into time, we have ^ = 20* 19* 4r.766. 

11. The angle POZy Fig. 1, between the vertical circle and 
the declination circle of a star, is frequently called the jxiraUactic 
angle y and will here be denoted by q. To find its value from the 
data ^, -4, and ^, we have the equations 

cos d cos ^ = sin C sin ^ + oos C cos f cos A \ ^i a 
cos d sin q = cos ip sin A ) 

which may be solved in the following form : 

/ sin F=:zf<\n ^ 

f cos F=^ cos C cos A 
cos d cos q =f cos (^ — F) 
cos d sin ^ = cos ^ sin A 

or in the following : 

g sin (r = sin 9 

g cos G -- cos ^ cos A 
cos d cos q = g cos (C — G) 
cos d sin q = cos ^ sin A 

or again in the following : 



* In this work the letter n prefixed to a logarithm indicates that the number to 
which it corresponds is to hare the negatiye sign. 


tan G = tan f sec A 
tan jr = 

tan A cos G v (13) 

COS (C — G) 

and, in the use of the last form, it is to be observed that q is to 
be taken greater or less than 180° according as ^ is greater or 
less than 180°, as is evident from the preceding forms. 

12. If, in a given latitude, the azimuth of a star of known 
declination is given, its hour angle and zenith distance may be 
found as follows. We have 

cos t sin ^ — sin t cot A = cos <p tan d 
cos C sin ^ — sin C cos tp cos A=^mi d 

The solution of the first of these is effected by the equations 

& sin J? = sin ^ 
h cos B = cot A 

sm (jB — ^) = ^ 

and that of the second by 

c sin C= sin <p 

c cos C= cos fp cos A 

sm (C — C) = 


13. Finally, if from the given altitude and azimuth we wish to 
find the declination, hour angle, and parallactic angle at the 
same time, it will be convenient to use Gauss's Equations, which 
for the triangle ABC, Fig. 3, are 

cos } a sin } (B + C) = cos } (6 — c) cos } A 
cos } a cos } (B + C ) = cos } (6 + c) sin J A . .^ 
sin } a sin } (B — C) = 8in } (6 — c)cos}A ' ^ ^ 
sin J a cos } (B — C) = sin } (6 + c) sin } A 

which are to be solved in the usual manner [Sph. Trig. Art. 
116] after substituting the values A = 180° — -4, 6 = C> <? = 
90° — ^, a = 90° — 5, B = /, C = q. 

14. Given the declination {8) and hour angle (t) of a staVy and the 
latitude (^), to find the zenith distance (^) and azimuth {A) of the star. 
That is, to transform the co-ordinates of the second system into 
those of the first. 

We take the same general equations (iE) of Spherical Trigo- 
nometry which have been employed in the solution of the pre- 


ceding problem, Art. 10; but we now suppose the letters A, B, 
C, in Fig. 3, to represent respectively the pole, the zenith, and 
the star, so that we substitute 

b = W — d B = 180<' — A 

c = 90^ — ^ 

and the equations become 

cos C = sin f sin d -\- cos f cos d cos t 
sin C cos A=z — cos ^ sin ^ -f- sin ^ cos d cos t y (14) 

sin C sin A = cos ^ sin t 

which express ^ and A directly in terms of the data. 
Adapting these for logarithmic computation, we have 

m sin M= sin d 

m cos M= cos 3 cos t 

cos Z =m cos (^ — M) ) (15) 

sin Z cos A = m sin (^ — M) 
sin C sin A = cos d sin t 

in which m is a positive number. 
Eliminating m, we deduce the following simple and accurate 

formulas : 

,-. tan^ 
tanif = 

cos t 

. tan t cos if . .^^. 

tan A = -7— 5^. ) (16) 

sin (vp — M) 

tan C = ^'^"^^-^) 

where A is to be taken greater or less than 180° according as t 
is greater or less than 180°. 

Example 1.— In latitude ^ = 38° 58' 53", there are given for 
a certain star, 5-= -8° 31' 46".5G, / = 20* 19- 41'.766; required 
A and ^. By (16) we have : 

log tan 6 fi0.l7A0310 

^ = 88« 68^ 53'^ log cos t 9. 7677077 log tan t wO. 1659995 

ir=— 14 40 48.98 log tan .V fi0^182«88 log cos if 9.9865859 

p^M ^ 5389^1.98 log tan (^— -V) 0.1333561 log co««c (^ — 1^)^0989172 

A=z 800 10 80 log cos A 9.70_12r)95 log tan A fiO.2356026 

C iss C9 42 80 log tan C 0.4320906 


For verification we can use the equation 

sin C sin A =: cos d sin t 

log sine 9.9721748 log cos ^ 9.9951697 

log sin A 9.9367621 log sin t 9.9137672 

9.9089369 9.9089369 

Example 2. — In latitude y>= —48^ 32', there are given for a 
star, 5 = 44° 6' 0", i = 17* 25"» 4* ; required A and C- 

We find A = 241° 53' 33".2, C = 126° 25' 6".6 ; the star is 
below the horizon, and its negative altitude, or depression, is 
A = — 86° 25' 6".6. 

K the zenith distance of the same star is to be frequently com- 
puted on the same night at a given place, it will be most readily 
done by the following method. In the first equation of (14) 


cos t = l — 2 sin' } t 
then we have 

cos C = cos {9-rd) — 2 cos f cos d sin" } t 

where ip>rd signifies either ip — d ord — <pj and if ^ > ^ the latter 

form is to be used. Subtracting both members from unity, we 


sin' } C = sin' J {9-^^) + cos f cos d sin' } t 
Now let 

m = i/cos ip cos d 

n = 8in } (sp*^^) 
then we have 

8inK = nJT7^5?^^ 
^ n' 

and hence, by taking an auxiliary N such that 

tan N= — sin J ^ 

we have ) (17) 

sin K = Ty=- — ^sin it 

cos iV Bin N 

The second form for sin \ f will be more precise than the first 
when sin iVis greater than cos N. 

The quantities m and n will be constant so long as the decli- 
nation does not vary. 

16. If the parallactic angle q (Art. 11) and the zenith distance 

Voi-L— 3 



j; an- rc-quired from the data y>, *, and <, they may be found 
fr^jiu the equations 

cos C = sin f sin d -f- cos f cos ^ cos t 
sio C cos ^f = sin ^ cos ^ — cos tp sin ^ cos < V (18) 

sin C sin ^ = cos ^ sin t 

which are adapted for logarithms as follows : 

n sin iV= cos ^ cos ^ 

ncosiV=8in ^ 
cos C = n sin (^ + iV) ^ (19) 

sin C cos q=zn cos \p + JV) 
sin C sin ^ = cos s^.sin t 

Off eliminating n, thus: 

tan JV=cot ^ cost 

. . tan ^ sin iV , 

tan C sin flr = \ (-20^ 

^ sin {d +N) ' ^ ^ 

tan C COH q =: cot (^ + iV^ 

VVIjirn this last fonu is employed in the case of a star which 
UiiH \n:i'U olmerved above the horizon, tan (^ is known to be posi- 
tive, and there is no ambiguity in the determination of q. This 
inrtn iH, tlierefore, tbe moHt convenient in practice. 

If f. A, and q are all rcciuired from the data 5, t, and ^, we 
luive, by (iauHn's eijuationH, 

Hin } C Hin J (A + q)=^ sin } ^ cos } (^ + d) 
sin i C COM i (A + q) -- com § ^ sin J (^ — d) 
iOH i C Hin } (/t — q) ~ sin it sin i (f + ^ 
<!OH i C eo8 1 (A — /jr) — cos M cos } (^p — d) 


fiK. 4. 

10. Wlii'n tbi! altitude, azimuth, and parallactic angle of known 

stars are to be frequently computed at the 
same place, the labor of computation is 
much diminished by an auxiliary table pre- 
jmred for the latitude of the place accord* 
ing to formulic proposed by Gauss. A 
Hpecimen of such a table computed for the 
latitude of the Altona Observatorv will be 
found in ^^&humacher's IlUlfstafcbiy neu 
lieniusg. V. AV'^arnstorif/* The requisite 
formnhe are readily deduced as follows: 

Let tli4» d<»< lination circle through the object 0, Fig. 4, be 
produced to intersect the horizon in F. By the diurnal motion 


the point F changes its position on the horizon with the time ; 
hut its position depends only on the time or the hour angle 
ZPO^ and not upon the declination of 0, The elements of the 
position of F may therefore be previously computed for succes- 
sive values of L 

We have in the triangle PFSj right-angled at Sj FPS==t, 
PS= 180° — ip; and if we put 

1Si = FSj J5 = P2?^— 90% y = 180° -P2?:a 
we find 

tan a = sin 9» tan t, tan B = cot f cos t, cot ^ = Bm B tan t 

We have now in the triangle HOF^ right-angled at H, 

B + d=OF, r = HFO, h = OS, 

and if we put 

u = SF = SS''FS = A--% 

we find 

tan u = cos T' tan (^B + d) A = ^ + u 

sin A = sin T' sin {B -f d) or, tan h = tan y sin u. 

To find the parallactic angle q=POZj we have in the triangle 

tan q = cot y sec (J? + d) 

In the Gaussian table for Altona as given in the "Hiilfstafeln" 
we find five columns, which give for the argument t, the quan- 
tities a. By log cos Yy log sin y, log cot y, the last three under 
the names log (7, log D, and log -B", respectively. With the aid 
of this table, then, the labor of finding any one of the quan- 
tities hy Ay q IB reduced to the addition of two logarithms, 

tan u= Ct&n (^B + d) sin h = D sin (B + d) 

A = a + t« tangr = j&8ec (B+d) 

The formulae for the inverse problem (of Art. 10) may also be 
found thus. Let G be the intersection of the equator and the 
vertical circle through 0, and put B = HGy u=DGy^=QGy 
Y = ZGQ; then we readily find 

tan 9i = sin f> tan A, tan B = cot ^ cos A, cot Y = QmB tan A 

which are of the same form as those given above, with the ex- 
change of A for L Hence the same table gives also the elements 
of the point Gy by entering with the argument "azimuth,** ex- 
pressed in time, instead of the hour angle. We then have ^ = 


DQ, and if we here put u = DG = ^ — t, we have from the 
triangle GDO 

sin J = sin ;* sin (h — B) tan u = cos y^ tan (h — B) 

or, employing the notation of the table, 

tan M = (7 tan (A — B) sin ^ = D sin (h — B) 

t = ^ — u tan ^ = J& see (A — B) 

17. To find the zenith distance and azimuth of a star^ when on the 
six hour circle. — Since in this case < = 6* = 90°, the triangle PZOy 
Fig. 4, is right-angled at P, and gives immediately 

cos ZO = cos PZ cos PO cot PZO = sin PZ cot PO 

or, since PZO = 180° — J, and cot PZO = - cot A, 

cos C = sin ^ sin d cot A= — cos ^ tan d 

But if the star is on the six hour circle east of the meridian, 

we must put t = 18*= 270° and PZ 0=^ — 180° ; hence for this 


cot A = -f cos ^ tan d 

A more general solution, however, is obtained from the equa- 
tions (14), by putting cos < = 0, sin < = ± 1, whence 

cos C = sin 9> sin d '\ 

sin C cos A = — cos ^ sin J V (22) 

sin C sin A = rb cos d ) 

the lower sign in the last equation being used when the star is 
east of the meridian. 

Example. — Required the zenith distance and azimuth of Sirius, 
8 = — 16° 31' 20", when on the six hour circle east of the meri- 
dian at the Cape of Good Hope, ^ = — 33° 56' 3". We find 

log (—cos 8) = log sin : sin A =n9.9816870 
log (— cos ^ sin d) = log sin C cos il = 9.3728204 

A = 283°49'84".9 
log sin A = 9.9872302 

log sin C = 9.9944568 
log sin 9> sin a = log cos C = 9 .2007309 

C =~80° 5r 55" 



Fig. 6. 

18. To find the hour avgle^ azimuihy and zenith distance of a given 
star at its greatest elongation. — In this case the vertical circle 
ZSj Fig. 5, is tangent to the diurnal circle, 
SAj of the star, and is, therefore, perpendicular 
to the declination circle PS. The right triangle 
PZS gives, therefore, 

tan ^ 

cos t = 

BmA = 

C08C = 

tan d 
cos 9 
cos ^ 
sin ^ 
sin S 


If d and <p are nearly equal, each of the quantities cos t, sin Ay 
and cos ^ will be nearly equal to unity, and a more accurate 
solution for that case will then be as follows : 

Subtract the square of each from unity ; then we have 

tan' d — tan* ^ sin (d -f ^) sin (d — ^) 

sin* t = 

eoB"il = 

Bin' C = 


cos' — cos* d 


sin'^ — sin'^ 

cos'^ sin'^ 

sin (d 4- 9) 81^ (^ — 9) 

sin (d + 9) sin (d — ^) 

sin' d sin' d 

Hence if we put 

k = y/[8in (d + 9») sin (d — ^)] 
we shall have 

k . k 

Bmt = 

COB A = 

sin C = 

sin 3 


cos ^ sin d cos ^ 

19. To find the hour angky zenith distance^ and parallactic angle of 
a given star on the prime vertical of a given place. 

In this case, the point in Fig. 1 being in the circle WZE^ 
the angle PZO is 90®, and the right triangle PZO gives 

tan d 

cos t = 

cosC = 

smgr = 

tan ^ 

sin d 
sin ^ 

cos 9> 
cos d 




If 5 is but little less than y>, the star will be near the zenith, 

and, as in the preceding article, we shall obtain a more accurate 

solution as follows : 


k = i/[8in (f + d) sin {<p — dj] 

Bin C = —, cos q = ^^^^ (26) 

sin t = 

sin ip cos d 



We may also deduce the following convenient and accurate 
formula for the case where the star's declination is nearly equal 
to the latitude [see Sph. Trig. Arts. 60, 61, 62] : 

tan ht =z 

tan } C = 

// sin (y — d) 
\ Vein (sp + d) 

I /tan J (sp — ^) 


tan } (f> + i)/ 
tan (45" — i q) = ^[tan i (f + S) tan } (^ 



If 5 > ^, these values become imaginary; that is, the star can- 
not cross the prime vertical. 

Example. — Required the hour angle and zenith distance of the 
star 12 Canum Venaticorum [d = -\- 39°»5' 20") when on the prime 
vertical of Cincinnati (^ = + 39° 5' 54"). 

^ — a = 0° 0' 34" 
^ + ^ = 78 11 14 

log sin {<p — d) 6.21705 
log sin lip + ^)_9.99070^ 


log tan } t 8.11318 

i f = 0° 44' 36".6 

t = 1° 29' 13".2 

= 0* 5- 56*.88 

}(^ — ^)= 0^0' 17" 
i (ip -^d)z= 39 5 37 

log tan i (^ —d) 5.91602 
log tan 1 (^ + ^) 9.90982 

log tan i C 8.00310 

i C = 0° 34' 37".3 
: = r 9'14".6 

20. To find the amplitude and hour angle of a given star ichen in 
the horizon. — If the star is at H, Fig. 1, we have in the triangle 
PILY, right-angled at N, PN ^ 9. HPN ^. 180° - U PH ^= 
90° — d\ and if the amplitude WH is denoted by a, we have 
HN= 90° — a. This triangle gives, therefore, 

sin a =1^ 8oc ^ sin d 
cos f =: — tan tp tan d 

} (28) 


21. Given the hour angle {t) of a siar^ to find its right ascension (a). 
— Transformation from our second system of co-ordinates to the 

There must evidently be given also the position of the meridian 
with reference to the origin of right ascensions. Suppose then 
in Fig. 1 we know the right ascension of the meridian, or VQ 
= 0, then we have VD = VQ —DQ, that is, 

a= e —t 

Conversely, if a and are Igiown, we have 

f = e — a 

The methods of finding at a given time will be considered 

22. Given the zenith distance of a knovm star at a given place, to 
find the star's hour angle^ azimuth^ and parallactic angle. 

In this case there are given in the triangle POZj Fig. 1, the 
three sides ZO = C, POP = 90° -8, PZ= 90° - ^, to find 
the angles ZPO = t, PZO = 180° — A, and POZ =q. The 
formula for computing an angle B of a spherical triangle ABC, 
whose sides are a, 6, c, is either 

sin J B = J / 8in(^-^)Bin(^-c) \ 
\ \ sin a sin <? / 

C08iB = ^/f «'"""'" (^-^) \ 
Af \ sin a sin c / 

or f^n^T^-- // 8in(5-a)sin(^-c) \ 

\ \ sin 5 sin (s — b) I 

in which 5 = J (a + 6 + <?)• We have then only to suppose B 
to represent one of the angles of our astronomical triangle, and 
to substitute the above corresponding values of the sides, to ob- 
tain the required solution. 

This substitution will be carried out hereafter in those cases 
where the problem is practically applied. 

23. Given the declination (8) and the right ascension (a) of a star, 
and the obliquity of the ecliptic (e), to find the latitude (fi) and the longi- 
tiule {X) of the star. — Transformation from the third system of co- 
ordinates to the fourth. 

The solution of this problem is similar to that of Art. 10. 



Fig. 6. 

The analogy of the two will be more apparent if we here repre- 
sent the sphere projected on the plane of 
the equator as in Fig. 6, where VBUCia 
the equator, P its pole ; VA U the ecliptic, 
P' its pole, and consequently CP'PB the 
solstitial colure; POD^ P'OL, circles of 
1^ declination and latitude drawn through the 
star 0. Since the angle which two great 
circles make with each other is equal to 
the angular distance of their poles, we have 
PP' = £ ; and since the angle P'PO is 

measured by CD and PP'Ohy AL^ we have in the triangle 


P'PO, PP'O, P'O, PO, PP' 

90^ + a, W" — X, 90<> — Py 90*» — ^, c 

which, substituted respectively for 





in the general equations (31), Art. 10, give 

sin /9 = cos e sin ^ — sin e cos d sin a 
cos ^ sin >l = sin e sin ^ -f- cos e cos d sin a 

cos P cos X = cos d cos a 


which are the required formulae of transformation. Adapting 
for loe^rithmic computation, we have 

m sin J[f = 
m cos M = 

sin /9 = 
cos fi B\n X =z 
cos fi cos >l = 

sin d 

cos d sin a 

m sin (M- 
m cos (M - 
cos S cos a 


in which m is a positive number. 

A still more convenient form is obtained by substituting 

k = 


cos d 

k' = 

cos ^ 


by which we find 



k sin M 
k cos JIf 
X/sin >l 
A:' cos >l 
tan fi 

cos fi sin X 

tan d 

sin a 

cos ( Jf — e) 

cos Jlif cot a 

sin X tan (Jlf 

cos (M — e) 


For verification : coB^Bina = coaJtf 

Example. — Given ^, a, and e as below, to find ^ and X. Com- 
putation by (31). 

a = — 16° 22' 35".45 
a = 6 33 29 .30 

e= 23 27 31.72 
log tan ^ = log k sin M n9.4681562 
log sin a = log A: cos M 9.0577093 

jf 3= _ 68*» 45' 41".87 
JM"— e = — 92 13 13 .59 

log cos M 9.5590070 
log cot a 0.9394396 

log A' cos >l 0.4984466 
logcos (-Sf — e) = log A-' sin >l n8.5882080 

X = 359*> 17' 43".91 

log sin >l W8.0897286 

log tan (ilf— e ) 1.4114658 

log tan ^ n9.5011944 

/5 = — 17° 35' 37".51 

log cos fi sin X n8.0689234 
log cos d sin a 9.0397224 
log o_08(Jtf-e) „9 02g20i(^ 


Tables for facilitating the above transformation, based upon 
the same method as that employed in Art. 16, are given in the 
American Ephemeris and Berlin Jahrbuch. The formulfe there 
used may be obtained from Fig. 6, in which the points F and G 
are used precisely as in Fig. 4 of Art. 16. 

24. K we denote the angle at the star, or P'OP, by 90° — Ey 
the solution of the preceding problem by Gauss's Equations is 

cos (45° 
cos (45° 
Bin (45° 
sin (45° 

i/9)sin HE—Xy 

: sin [45° 
cos [45° 
sin [45° 
cos [45° 

i (e + ^)] cos (45° + } a) 

} (e + d)] COS (45° + J a) 

J(e — ^)]sin(45° + Ja) 


25. If the angle at the star is required when the Gaussian 
Equations have not been employed, we have from the triangle 
jPOP', Fig. 6, putting P'OP = 7i = W — JS, 


COS /5 COS Tj = COS e COS ^ -|- sin € sin d sin a 
COS ^ sin 17 ^ sin e cos a 

or, adapted for logarithms, 

n sin iV=ir sin e sin a 

n cos iV = cos e 
cos ^ cos ly = n cos (N—d) ( (^) 

cos fi Bin Tj =z sin e cos a 

26. Given the latitude (j9) and longitude {X) of a star^ and the 
obliquity of the, ecliptic (e), to find the declination and right ascension 
of the star. 

By the process already employed, we derive from the triangle 
FP'O, Fig. 6, for this case, 

sin d = cos e sin ^ + sin e cos fi sin X 
cos ^ sin a = — sin € sin ^ -|- cos e cos ^ sin A )- (34) 

cos 6 cos a = cos fi cos >l 

which, it will be obsen-ed, may be obtained from (29) by inter- 
changing a with Xy and 8 with ^, and at the same time changing 
the sign of e, that is, putting — e for e, and, consequently, — sin e 
for sin e. 
For logarithmic computation, we have 

m sin JW= sin ^ 

m cos M = cos ^ sin X 

sin d = m sin (3/ + c) ) (36) 

cos ^ sin a = m cos {M + e) 

cos d cos a = COS /5 COS X 

or the following, analogous to (31) : 

A* sin M = tan ff 
k cos ^f= sin >l 
k' sin a = cos (M + c) 
A'' cos a =r-. CO8 Jf cot A 
tan d -- sin a tan (-If + t) 

__ . , . coH r) sin a cos (^f -f c") 

For verification : k: •— t= — . " ir — 

-^ cos /Sf sm >l cos M 


27. The angle at the star, POP', being denoted, as in Art. 24, 


by 90° — Ey the solutiou of this problem by the Gaussian 
Equations is 

sin (45<^— J d) sin \(JE+d) = sin [45°— i (e + ^)] sin (45°+ J X) 
sin (45°— J a) cos } (J5: + a) = cos [45°— } (e — ^5)] cos (45°+ } X) 
cos (45°— i d) sin } (jE — a) = cos [45°— } (e — /9)] sin (45°+ i A) 
cos (45°— } d) cos i (^ — a) = sin [45°— } (e + /9)] cos (45°+ } X) 


28. But if the angle yj = 90° — j& is required when the 
Gaussian Equations have not been employed, we have directly 

cos d cos Ti = cos € cos fi — sin s sin fi sin X 
cos ^ sin 17 = sin e cos X 

or, adapted for logarithms, 

n sin JV= sin c sin >l 

n cos iV= cos c 
cos ^ cos ly = n cos (iV + ^) [ (^^) 

cos ^ sin ly == sin € cos X 

29. i^or <A€ sun, we may, except when extreme precision is 
desired, put /3 = 0, and the preceding formulae then assume very 
simple forms. Thus, if in (34) we put sin /3 = 0, cos /? = 1, we 

sin ^ = sin e sin X 
cos ^ sin a = cos e sin X 

cos d cos a = COS X 

whence if any two of the four quantities 5, a, ^, e are given, we 
can deduce the other two. 


30. By means of spherical co-ordinates we have expressed 
only a star's direction. To define its position in space com- 
pletely, another element is necessary, namely, its distance. In 
Spherical Astronomy we consider this element of distance only 
so far as may be necessary in determining the changes of 
apparent direction of a star resulting from a change in the point 
from which it is viewed. For this purpose the rectangular co- 
ordinates of analytical geometry may be employed. 

Three planes of reference are taken at right angles to each 
other, their common intersection, or origin, being the point of 


obserration; and the star's distances from these planes are 
denoted by x^ y, and z respectively. These co-ordinates are 
respectively parallel to the three axes (or mutual intersections 
of the planes, taken tvvo and two), and hence these axes are 
called, respectively, the axis of x, the axis of y, and the axis of z. 
The planes are distinguished by the axes they contain, as "the 
plane of xy,'' the "plane of xz," the "plane of y^/' The co- 
ordinates may be conceived to be measured on the axes to 
which they belong, from the origin, in two opposite directions, 
distinguished by the algebraic signs of Tplus and minus^ so that 
the numerical values of the co-ordinates of a star, together with 
their algebraic signs, fully determine the position of the star in 
space without ambiguity. 

Of the eight solid angles formed by the planes of reference, 
that in which a star is placed will always be known by the signs 
of the three co-ordinates, and in one only of these angles will 
the three signs all be plus. This angle is usually called the first 
angle. To simplify the investigations of a problem, we may, if 
we choose, assume all the points considered to lie in the first 
angle, and then treat the equations obtained for this simplest 
case as entirely general; for, by the principles of analytical 
geometry, negative values of the co-ordinates which satisfy such 
equations also satisfy a geometrical construction in which these 
co-ordinates are drawn in the negative direction. 

The polxir co-ordinates of analytical geometry (of three dimen- 
sions) when applied to astronomy are nothing more than the 
Hphcrical co-ordinates we have already treated of, combined 
with the element distance ; and the formuhe of transformation 
from rectangular to polar co-ordinates are nothing more than 
the values of the rectangular co-ordinates in terms of the dis- 
tance and the spherical co-ordinates. For the convenience of 
reference, we whall here recapitulate these fornmla?, with special 
reference to our several systems of spherical co-ordinates. 

31. We shall find it useful to premise the following 
Lemma. — The distanre of a point in space from the plane of ant/ 
great circle of the celestial sphere is equal to its distance from the centre 
of the sphere multiplied by the cosine of its angular distance from the 
pole of that circle; and its distance from the axis of the circle is equal to 
its distanre from the centre of the sphere multiplied by the sine of its 
angular distance from the pole. 


For, let ABy Fig. 7, be the given great circle orthographi- 
cally projected upon a plane passing through its axis OP and 
the given point C; P its pole. The dis- 
tance of the point C from the plane of the 
great circle is the perpendicular CD ; CE 
is its distance from the axis; CO its dis- 
tance from the centre of the sphere; and 
the angle COP the angular distance from 
the pole. The truth of the Lemma is, 
therefore, obvious from the figure. 


82. The values of the rectangular co-ordinates in our several 
systems may be found as follows : 

First system. — Altitude and azimuth. — ^Let the primitive plane, 
or that of the horizon, be the plane of xy; that of the meridian, 
the plane of xz; that of the prime vertical, the plane of yz. 
The meridian line is then the axis of x; the east and west line, 
the axis of y; and the vertical line, the axis of z. Positive x 
will be reckoned from the origin towards the south, positive y 
towards the west, and positive z towards the zenith. The first 
angkj or angle of positive values, is therefore the southwest 
quarter of the hemisphere above the plane of the horizon. Let 
Z, Fig. 8, be the zenith, S the south point, W the Fig. 8. 

west point of the horizon. These points are 
respectively the poles of the three great circles 
of reference ; if, then, A is the position of a 
star on the surface of the sphere as seen from 
the centre of the earth, and if we put 

h = altitude of the star = AH, 
A = azimuth « = SHy 

J = its distance from the centre of the sphere 

we have immediately, by the preceding Lemma, 

X = J cos ASf y = J cos AWy z = A cos AZ^ 
which, by considering the right triangles AHSj AHWj become 

a: =r J cos h co8 A \ 

y = A cos A sin A > (39) 

j2r =: J sin i^ J 

These equations determine the rectangular co-ordinates x^y^z. 


when the polar co-ordinates J, A, A are given. Conversely, if 
Xy y, and z are given, we may find J, A, and A; for the first two 
equations give 

tan A = — 


and then we have 

J sin A = 2: 

J cos A = — ^ — = U. 

cos A sin A 

whence A and A. Or, by adding the squares of the first two 
equations, we have 

J cos h = y^x* + y' 

tan h = 

and the sum of the squares of the three equations gives 

J == y/^x" + y« + 2^) 

Second system. — Declination and hour angle. — ^Let the plane of 
the equator be the plane of xy; that of the meridian, the plane 
of xz; that of the six hour circle, the plane of i/z. In the pre- 
ceding figure, let Z now denote the north pole, S that point of 
the equator which is on the meridian above the horizon and 
from which hour angles are reckoned, TFthe west point. Posi- 
tive X \vi\\ be reckoned towards *§, positive y towards the west, 
positive z towards the north. If then A is the place of a star on 
the sphere as seen from the centre, and we put 

d =-. the star's declination = Aff, 
t = " hour angle = SH, 

A= " distance from the centre, 

and denote the rectangular co-ordinates in this case by x\ y', z'^ 

we have 

a/ rT= J cos d cos t 

y' = J cos ^ sin e y (40) 

y = J sin ^ 

Third system, — Declination and right ascension. — Let the plane 
of the equator be the phme of xy; that of the equinoctial colure, 
the plane of xz; that of the solstitial colure, the plane of yz. 


The axis of x is the intersection of the planes of the equator 
and equinoctial colure, positive towards the vernal equinox ; the 
axis of y is the intersection of the planes of the equator and sol- 
stitial colure, positive towards that point whose right ascension 
is +90° ; and the axis of z is the axis of the equator, positive 
towards the north. K then, in Fig. 8, Z is the north pole, W 
the vernal equinox, A a star in the first angle, projected upon 
the celestial sphere, and we put 

H = doclination of the star = AH, 
a = right ascension " = WS, 
A = distance from the centre, 

while z'\ y"y z" denote the rectangular co-ordinates, we have 
af'= J cos ATF, y"= J cos AS, 2^' = A cos AZ, 

which become 

od' = A cos d cos a ^ 

y" = J cos ^ sin a > (41) 

2^' = A sin d ) 

Fourth system. — Celestial latitude and longitude. — ^Let the plane 
of the ecliptic be the plane of xy ; the plane of the circle of 
latitude passing through the equinoctial points, the plane of xz ; 
the plane of the circle of latitude passing through the solstitial 
points, the plane of yz. The positive axis of x is here also the 
straight line from the centre towards the vernal equinox ; the 
positive axis of y is the straight line from the centre towards the 
north solstitial point, or that whose longitude is +90° ; and the 
positive axis of z is the straight line from the centre towards 
the north pole of the ecliptic. 

If then, in Fig. 8, Z now denotes the north pole of the ecliptic, 
W the vernal equinox, A the star's place on the sphere, and 
we put 

/9 = latitude of the star = AHy 

X = longitude of the star = WS, 

A = distance of the star from the centre, 

and x'", 2/'", z'", denote the rectangular co-ordinates for this 

system, we have 

a/" = A cos /5 cos X 

y"' = J cos /9 sin X } (42) 

y" = J sin /9 




33. For the purposes of Spherical Astronomy, only the most 
simple cases of the general transformations treated of in analy- 
tical geometry are necessary. We mostly consider but two cases: 

First. Transformation of rectangular co-ordinates to a new originj 
icithout changing the system of spherical co-ordinates. 

The general planes of reference which have been used in this 
chapter may be supposed to be drawn through any point in space 
without changing their directions, and therefore without changing 
the great circles of the infinite celestial sphere which repre- 
sent them. We thus repeat the same system of spherical co-ordi- 
nates with various origins and difterent systems of rectangular 
co-ordinates, the planes of reference, however, remaining always 
parallel to the planes of the primitive system. 

The transformation from one system of rectangular co-ordi- 
nates to a parallel system is evidently etfected by the formulae 

x^ = Xj, -f- a 


in which rp y,, z^^ are the co-ordinates of a point in the primitive 
system ; r,, y,, z^ the co-ordinates of the same point in the new 
system ; and cr, 6, c are the co-ordinates of the new origin taken 
in the first system. 

As we have shown how to exj^ress the values of x^^ y^ z^ and 
of Tj, 2/2, ^2 "^ terms of the spherical co-ordinates, we have only 
to substitute these values in the preceding formula? to obtain the 
general relations between the spherical co-ordinates correspond- 
ing to the two origins. This is, indeed, the most general method 
of determining the eftect of parallax^ as will appear hereafter. 

Second. Transformation of rectangular co^ 
ordinates tchen the system of spherical co-ordi- 
nates is changed but the origin is unchanged. 
This amounts to changing the directions of 
the axes. The cases which occur in practice 
are chiefly those in which the two systems 
^ have one plane in common. Suppose this 
plane is that of xz, and let OT, OZ, Fig. 9, be 
the axes of z and z iu the first system; 0-^^, 

Fig. y. 


OZ^J the axes of Xj and z^ in the new system. Let A be the 
projection of a point in space upon the common plane; and 
let z = ABy 2 = OBy x^=AB^y z^=OBy The distance of the 
point from the common plane being unchanged, we have 


The axis of ^ may be regarded as an axis of revolution about 
which the planes of yx and yz revolve in passing from the first to 
the second system ; and if u denotes the angular measure of this 
revolution, or u = XOX^ = ZOZ^ = BAB^^ we readily derive 
from the figure the equation 

X sec u=^x^ — 2j tan u 

or, multiplying by cos u, 

x = Xi C08M — z^ sin u 

2 =x tan u-\-2i sec u 

or, substituting in this the preceding value of a:, 

2 =zx^H\n u-^-z^ cos u 

Thus, to pass from the first to the second system, we have the 


x = x^cosu — z^Binu '\ 

y = yi > (44) 

2r = Xj sin w + z^ cos u ) 

And to pass from the second to the first, we obtain with the same 


X^= X cos U-^-ZBlUU 

yt= y }- (45) 

2^ = — iC sin M + zcos u 

As an example, let us apply these to transforming from our 
second system of spherical qo-ordinates to the first ; that is, from 
declination and hour angle to altitude and azimuth. The meri- 
dian is the common plane ; the axis of z in the system of declina- 
tion and hour angle is the axis of the equator, and the axis of ^i 
in the system of altitude and azimuth is the vertical line ; the 
angle between these axes is the complement of the latitude, or 
u = 90® — <p. Substituting this value of u in (44), and also the 
values of a:, y, ^, Xj, y,, ^„ given by (39) and (40), we have, after 
omitting the common factor J, 

Vol. L— 4 


COS li COS A=^9\n fp COS $ cos t — cos f sin H 
cos A sin -^ = cos d sin t 

sin h = cos fp cos ^ cos t + sin tp sin ^ 

which agree with (14). We see that when the element of dis- 
tance is left out of view (as it must necessarily be when the 
origin is not changed), the transformation by means of rectangu- 
lar co-ordinates leads to the same forms as the direct application 
of Spherical Trigonometry. . With regard to the entire generality 
of these formulae in their application to angles of all possible 
magnitudes, see Sph. Trig. Chap. IV. 


34. It is often necessary in practical astronomy to determine 
what effect given variations of the data will produce in the quan- 
tities computed from them. Where the formulre of computa- 
tion are derived directly from a spherical triangle, we can employ 
for this purpose the equations of finite diffvrcnccs [Sph. Trig. 
Chap. VI.] if we wish to obtain rigorously exact relations, or 
the simpler differential equations if the variations considered 
are extremely small. As the latter case is very frequent, I shall 
deduce here the most useful differential formuhe, assuming as 
well known the fundamental ones [Sph. Trig. Art. 153], 

da — cos C dh — cos B r/c trrr 8in 6 sin C d\ ^ 

— cos C da 4- dh — cos Kdc ^=z sin c sin A ^B > (46) 

— cos B tia — cos A db -\- dc — . Hin a sin B dC ) 

From these we obtain the following by eliminating da: 

sin db — cos a sin B dc ^^ sin b cos C d\ + sin a dB ) ,^-.x 
— cos rt sin C db + sin B dc —Hinc cos B d\ -f sin a dC ) 

and by eliminating db from these: 

sin a sin B dc = cos b d\ + cos a dB + dC (48) 

If we eliminate d\ from (47), we find 

cos b sin C db — cos c sin B dc -— sin c cos B dB — sin h cos C dC 

the terms of which being divided either by sin b sin C, or by its 
equivalent sin c sin H, we obtain 

cot b db — cot c dc =-- cot B (/B — cot C dC (49) 


35. As an example, take the spherical triangle formed by the 
zenith, the pole, and a star, Art. 10, and put 

A = 180^ — ^ a = 90° — ^ 

C = ^ c = 90** — sp 

then the first equations of (46) and (47) give 

dd = — cos q dZ -{- sin q sin C dA -f- cos t d^ ^ x^qx 

cos d dt = sin q dZ -{- cos q sin C dA -f- sin i sin t d<p ) 

which determine the errors dd and dt in the values of d and t 
computed according to the formulae (4), (5), and (6), when (^, A^ 
and <p are affected by the small errors d^, dA, and d<p respectively. 
In a similar manner we obtain 

c?C = — cos q dd -{- sin q cos d dt -}- cos Ad^ 1 .;.j. 

sin C </^l = sin q dd -j- cos ^ cos d dt — cos C sin Ad<p ) 

which determine the errors rfj and dA in the values of f and A 
computed by (14), when 5, t, and ^ are aflfected by the small 
errors dd, dt, and dip respectively. 

36. As a second example, take the triangle formed by the pole 
of the equator, the pole of the ecliptic, and a star. Art. 23. De- 
noting the angle at the star by tj, we find 

d^ = cos Tj dd — sin j) cos d da — ein X de ) .^«,. 

cos fi dX = sin y^ dd -{■ cos rj cos d da -{- sin fi cos X ds j v *-; 

and reciprocally, 

dd = cos ij di3 -f- sin rj cos /9 dX -[- sin a <7e ) ,-qv 

cos d da = — sin ly di3 -(- cos ly cos ^ c?>l — sin d cos a <fe J ^ 

52 TiUE. 




37. Transit. — The iiifltant when any point of the celestial 
nphere is on the meridian of an observer is designated as the 
tramii of tliat point over the meridian; also the meridian jxiasage^ 
and culmi)iaiio)i. In one complete revolution of the spliere 
about its axis, every point of it is twice on the meridian, at 
points which are 180° distant in right ascension. It is therefore 
necessarv to distin«:uish between the two transits. The meri- 
dian is bisected at the poles of the ccpuitor: the transit over that 
half of the meridian whi(»h contains the ol)scr\'er'8 zenith is the 
upper transit, or culmination; that over the half of the meri- 
dian which contains the nadir is the lower transit, or culmina- 
tion. At the upper transit of a point its hour angle is zero, or 
0* ; at the lower transit, its hour angle is 12\ 

38. The motion of the earth about its axis is perfectly uni- 
fona. If, then, the axis of the earth i)rescrved precisely the 
same direction in space, the ai»parcnt diurnal motion of the 
celestial sphere would also be perfectly uniform, and the inter- 
vals between the successive transits of any assumed point of the 
sphere would be perfectly equal. The effect of changes in the 
position of the earth's axis upon the transit of stai-s is most per- 
ceptible in the case of stars near the vanishing points of the 
axis, that is, near the poles of the heavens. AVe obtain a measure 
of time srn^ihf^ uniform by employing the successive transits of 
a point of the ecpnitor. The point most naturally indicated is 
the nrnal ctpunojr (also <*alled the Fii*st point of Aries, and de- 
noted by the symbol for Aries, T). 

30. A siiUrenl ilmj is the interval of time between two succes- 
sive (upper) transits of the true vernal ecpiinox over the same 

The ettect of preiession and nutation upon the time of transit 

TIME. 53 

of the vernal equinox is so nearly the same at two successive 
transits, that sidereal days thus defined are sensibly equal. (See 
Chapter XI. Art. 411.) 

The sidereal time at any instant is the hour angle of the vernal 
equinox at that instant, reckoned from the meridian westward 
from 0* to 24*. 

When TP is on the meridian, the sidereal time is 0* 0* 0* ; and 
this instant is sometimes called sidereal noon. 

40. A solar day is the interval of time between two successive 
upper transits of the sun over the same meridian. 

The soUir time at any instant is the hour angle of the sun at 
that instant. 

In consequence of the earth's motion about the sun from west 
to east, the sun appears to have a like motion among the stars, 
or to be constantly increasing its right ascension ; and hence a 
solar day is longer than a sidereal day. 

41. Apparent and mean solar time. — If the sun changed its right 
ascension uniformly, solar days, though not equal to sidereal days, 
would still be equal to each other. But the sun*8 motion in right 
ascension is not uniform, and this for two reasons : 

1st. The sun does not move in the equator, but in the ecliptic, 
BO that, even were the sun's motion in the ecliptic uniform, its 
equal changes of longitude would not produce equal changes of 
right ascension; 2d. The sun's motion in the ecliptic is not uni- 

To obtam a uniform measure of time depending on the sun's 
motion, the following method is adopted. A fictitious sun, which 
we shall call the/r6'^ mean sun^ is supposed to move uniformly at 
such a rate as to return to the perigee at the same time with the 
true sun. Another fictitious sun, which we shall call the second 
mean sun (and which is often called simply the mean sun), is sup- 
posed to move uniformly in the equator at the same rate as the 
first mean sun in the ecliptic, and to return to the vernal equinox 
at the same time with it. Then the time denoted by this second 
mean sun is perfectly uniform in its increase, and is called mean time. 

The time which is denoted by the true sun is called the tnie 
or, more commonly, the apparent time. 

The instant of transit of the true sun is called apparent noon, and 
the instant of transit of the second mean sun is called ynean noon. 



Tlie (quation of time is the difFerence between apparent and 
r.'j'.an time ; or, in other words, it is the difterence between the 
hour angles of the trne sun and the second mean sun. The 
; difFerence is about IG" 

The equation of time is also the difterence between the right 
a-''*;iiHioiis of the true sun and the second mean sun. The right 
a-rension of the second mean sun is, according to the preceding 
'h'fmitionH, equal to the longitude of the liret mean sun, or, as it 
i- 'ommonly called, the sun*s mean longitude. To compute the 
«'qiiatioii of time, therefore, we must know how to iind the longi- 
VuU: of the first mean sun ; and this'is deduced from a knowledge 
of the true sun's apparent motion in the ecliptic, which belongs 
to Phyrti<;al Astronomy. Here it suffices us that its value 18 
iriven for each day of the year in the Ephemeris, or Nautical 

42. Asfronomkal time. — The solar day (apparent or mean) is 
concreived by astronomers to commence at noon (apparent or 
nn'un), and is divided into twenty-four hours, numbered succes- 
nivclv from to 24. 

Astronomical time (apparent or mean) is, then, the hour angle 
of the sun (apjmrent or mean), reckoned on the equator xce^t- 
ward throughout its entire circumference from 0* to 24*. 

43. Cirll time. — For the conmion pur[)oses of life, it is more 
convenient to begin the day at midnight, that is, when the sun 
is on the meridian at its lower transit 

The civil (hiy is divided into two periods of twelve hours each, 
namely, from midnight to noon, marked A.M. (Ante Meridiem), 
and from noon to midnight, marked P.M. (Post Meridiem) 

44. T(i convert eiril into astronomical tirne, — The civil day begins 
12'^ before the astronomical day of the same date. This remark 
is the only precept that need be given for the conversion of one 
of these kinds of time into the other. 


Ast. T. May 10, ir)*=_r Civ. T May 11. 3* A.M. 
Ast. T. Jan. 3, 7*.-: Civ. T. Jan. 8, 7* P.M. 
Ast. T. Aug 31, 20* r-_ Civ. T. Sept. 1, 8» A.M. 

TIME. 55 

45. Time at different meridians. — The hour angle of the sun at 
any meridian is called the local (solar) time at that meridian. 

The hour angle of the sun at the Greenwich meridian at the 
same instant is the corresponding Greenwich time. This time we 
shall have constant occasion to use, both because longitudes 
in this country and England are reckoned from the Greenwich 
meridian, and because the American and British Nautical 
Almanacs are computed for Greenwich time.* 

The* difference between the local time at any meridian and the 
Greenwich time is equal to the longitude of that meridian from 
Greenwich, expressed in time, observing that 1* =^ 15°. 

The difterence between the local times of any tvvo ^ig- ^o- 
meridians is equal to the difterence of longitude of 
those meridians. 

In comparing the corresponding times at two dif- 
ferent meridians, the most easterly meridian may be 
distinguished as that at which the time is greatest ; 
that is, latest. ^ m a m^ 

If then PM^ Fig. 10, is any meridian (referred to the celestial 
sphere), PG the Greenwich meridian, PS the declination circle 
tlirough the sun, and if we put 

!r„ = the Greenwich time = GPS, 

T =z the local time = MPS, 

L = the west longitude of the meridian PM = GPMj 

we have 

T^=T^L I y^^^ 

If the given meridian were east of Greenwich, as PJM^', we 
should have its east longitude = T — T^\ but we prefer to use 
the general formula Z/= T^— T in all cases, observing that east 
longitudes are to be regarded as negative. 

In the formula (54), T^ and T are supposed to be reckoned 
always westward from their respective meridians, and from 0* to 
24* ; that is, T^ and T are the astronomical timeSy which should, of 
course, be used in all astronomical computations. 

As in almost every computation of practical astronomy we are 
dependent for some of the data upon the ephemcris, — and these 

♦ What we have to say respecting the Greenwich time is, however, equally appli- 
cable to the time at any other meridian for which the ephemeris may be computed. 

56 TIME. 

are commonly given for Greenwich, — it is generally the first step 
in such a computation to deduce an exact or, at least, an ap- 
proximate value of the Greenwich astronomical time. It need 
hardly be added that the Greenwich time should never be other- 
wise expressed than astronomically.* 


1. In Longitude 76° 32' W. the local time is 1856 April 1, 
9* 3'^ lO- A.M. ; what is the Greenwich time ? 

Local Ast. T. March 31, 21» 3- 10- 
Longitude +56 8 

Greenwich T. -Aprin^ 2 9 18 

2. In Long. 105° 15' E. the local time is August 21, 4* 3" P.M ; 
what is the Greenwich time ? 

Local Ast. T. Aug. 21, 4» 3* 
Longitude — 7 1 

Greenwich T. Aug. 20, 21 2 

3. Long. 175° 30' W. Loc. T. Sept. 30, 8* 10- A.M. = G. T. 
Sept. 30, 7* 52-. 

4. Long. 165° 0' E. Loc. T. Feb. 1, 7* 11- T.M. = G. T. Jan. 
31, 20* 11-. 

5. Long. 64° 30' E. Loc. T. June 1, 0* M. (Noon) n= G. T. May 
31, 19* 42-. 

46. In nautical practice the obsers'cr is provided with a chro- 
nometer which is regulated to Greenwich time, before sailing, 
at a phice wliose longitude is well known. Its error on Green- 
wich time is carefully determined, as well as its daily gain or 
loss, that is, its rafc, so that at any subsecpient time the Green- 
wich time niavbe known from the indication of the chronometer 
corrected for its error and the accumulated rate since the date 
of sailing. As, however, the chronometer has usually only 12* 
marked <m the dial, it is necessary to distinguish whether it 
indicates A.M. or I\M. at (ireenwich. This is ahvavs readilv 
done by means of the observer's approximate longitude and local 

* On this account, chronometern intended for nautical and astronomical purposet 
Mhould always be marked from <** to 24*. instead of from (»* to 12* as is now usual. 
It id 8urj>ri!«ing that navigators have not insisted upon this point. 

TIME. 57 

time. As this is a daily operation at sea, it may be well to illus- 
trate it by a few examples. 

1. In the approximate longitude 5* W. about S^ P.M. on Au- 
gust 3, the Greenwich Chronometer marks 8* ll*" 7*, and is fast 
on G. T. 6"* 10' ; what is the Greenwich astronomical time ? 

Approx. Local T. Aug. 3, 3» Gr. Chronom. 8* ll* 7* 

" Longitude, + 5 Correction, — 6 10 

Approx. G. T. Aug. 3, 8 Gr. Ast. T. Aug. 3, 8 4 57 

2. Li Long. 10* E. about 1* A.M. on Dec. 7, the Greenwich 
Chronometer marks 3* 14"* 13'.5, and is fast 25** 18'.7 ; what is 
theG. T.? 

Approx. Local T. Dec. 6, 13» Gr. Chronom. 3* 14- 13'.5 

<* Long. — 10 Correction, — 25 18 .7 

Approx. G. T. Dec. 6, 3^ G. A. T. Dec. 6, 2 48 54.8 

3. In Long. 9* 12" W. about 2* A.M. on Feb. 13, the Gr. Chron. 
marks 10* 27"* 13*.3, and is slow 30- 30^.3; what is the G. T.? 

Approx. Local T. Feb. 12, 14* Gr. Chronom. 10» 37- 13'.3 
" Long. +9_ Correction, + 30 30.3 

Approx. G. T. Feb. 12, ~23^ G. A. T. Feb. 12, 23 7 43-6 

The computation of the approxim'ate Greenwich time may, of 
course, be performed mentally. 

47. The formula (54), i= 7;— T, is true not only when T^ 
and T are solar times, but also when they are any kinds of time 
whatever, or, in general, when 7J, and 7^ express the hour angles 
of any point whatever of the sphere at the two meridians whose 
difterence of longitude is L. This is evident from Fig. 10, where 
S may be any point of the sphere. 

48. To convert the apparent time at a given meridian into the mean 
iimCj or the mean into the apparent time. 

If M = the mean time, 

A = the corresponding apparent time, 
E = the equation of time, 
we have 

M=A + lf! 

or A =M--E 

08 TIME. 

in which E is to l>c regarded as a positive quantity when it is 
adffifici to ajfpurent t'uiu\ The vahie of E is to be taken from the 
Xautical Ahuanae for the Greenwich instant corresponding to 
the given h)cal time. If apparent time is given, find the Gr. 
apparent time and take E from page I of the month in the 
Xautical Ahuanae; if mean time is given, find the Gr. mean 
time and take E from page II of the month. 

Example 1.— In h^ngitude 60° W., 1856 May 24, 3* 12* IQr 
P.M., apparent time; what is the mean time? 
We have first 

Local time May 2i, 3* 12- 10' 
Longitude, 4 

Gr. app. time May 24, 7 12 10 

We must, therefore, find E for the Gr. time. May 24, 7* 12* 
10', or 7\21. By the Xautical Almanac for 18r)6, we have ^at 
ajiparcnt (ireenwich noon May 24 - - 8"*25'.43, and the hourly 
dittercnce -j- 0'.224. Hence at the given time 

/; :=.. — 3- 2;V.43 + 0'.224 X 7.21 =:= — 3- 23*.81 

and the recpiired mean time is 

M. - 3* 12- 10- — 3- 23'.«1 ~. 3* 8- 40'. 19. 

Example 2.— In longitude 60° W., 18;56 May 24, 3*8-46M9 
mean time; what is the api>arcnt time? 

(Jr. mean time. May 24, 7* S<- 46-11) ( - 7M5) 

/; at mean noon May 24 =-- — 3- 2;V.41 Hourly diff. — 0'.224 

Correction for 7*. IT) — 4- 1 .r»0 7.15 

E=^— 3 23.S1 im 

and hence 

— A'- :+ 3 2:i .HI 
A -^:i 12 10 .00 

As tlM* equation of time is not a uniformly varying quantity, it 
is not (juite arcuratc to eomiJUtc its correction as above, hy mnl- 
tii»Ivinir the ixiviMi honrlv <liffercncc hv the numher of hours in 
the (Jreenwieli tinn*. ft»r tliat ]»rocess assumes that this hourly 
differen«M' is the same for eaeh hour. The variations in the 
h«»urly ditlerence are, however, so small that it is oidy when 

TIME. 59 

extreme precision is required that recourse must be had to the 
more exact method of interpolation which will be given here- 

49. To determine the relative length of the solar and sidereal \inits 
of time. 

According to Bessel, the length of the tropical year (which is 
the interval between two successive passages of the sun through 
the mean vernal equinox) is 365.24222 mean solar days;* and 
since in this time the mean sun has described the whole arc of 
the equator included between the two positions of the equinox, 
it has made one transit less over any given meridian than the 
vernal equinox ; so that we have 

366.24222 sidereal days = 365.24222 mean solar days 
whence we deduce 

1 Bid. day = ^^^'^^^^^ sol. day =' 0.99726957 sol. day 
^ 366.24222 ^ ^ 


24* sid. time = 23* 56* 4-.091 solar time 


1 sol. day = ^^j^ gi^. day = 1.00273791 sid. day 
^ 365.24222 ^ ^ 


24* sol. time == 24* 3- 56«.555 sid. time 

If we put 

366.24222 ^^ 


and denote by /an interval of mean solar time, by /' the equiva- 
lent mterval of sidereal time, we always have 

/' = /i7 = /-I- 0£ — 1) 7 =7+ .00273791 7 \ 

I =i^ =r^n — l)r = 7' — .00273043 r } ^^^^ 
ft ^ fi^ ) 

Tables are given in the Nautical Almanacs to save the labor of 
computing these equations. In some of these tables, for each 
solar inter\^al 7 there is given the equivalent sidereal interval 
I' = fiT, and reciprocally: in others there are given the correc- 
tion to be added to 7 to find 7' {i.e. the correction .00273791 7), 

* The length of the tropical year is not absolutely constant. The value given in 
the text is for the year 1800. Its decrease in 100 years is about C'.6 (Art. 407). 


and the correction to he subtracted from /' to find / (i.e. the 
correction .00273043 /'). The latter form is the most conve- 
nient, and is adopted in the American Ephemeris. The correction 
(/i — 1) 7 is frequently called the accckration of the fixed stars (re- 
latively to the sun). The daily acceleration is 3"* 56'.665. 

50. To convert the mean solar time at a givai meridian into the 
corresponding sidereal time. 

In Fig. 1, page 25, if PQ is the given meridian, VQthe equator, 
D the mean sun, Vthe vernal equinox, and if we put 

T= I)Q=i the moan solar time, 
0= VQ = the sidereal lime, 

:= the right ascension of the meridian, 
V= the right ascension of tho~mean sun, 

we have 

e==T+V (56) 

The right ascension of the mean sun, or V, is given in the 
American Ephemeris, on page II of the month, for each Green- 
wich mean noon. It is, however, there called the " Sidereal 
Time,*' because at mean noon the second mean stin is on the 
meridian, and its right ascension is also the right ascension of 
the meridian, or the sidereal time. But this quantity V is uni- 
formly increasing* at the rate of 3"* 56'.555 in 24 mean solar 
hours, or of i>*.8565 in one mean hour. To find its value at the 
given time T, we may first find the Greenwich mean time T^ by 
applying the longitude ; then, if we put 

T'i, ^n the value of V at Gr. mean noon, 

= the " sidereal tinio" in the ephemeris for the given date, 

we have 

in which T^ must be expressed in hours and decimal parts. It 
is easilv seen that 1>'.8565 is the acceleration of sidereal time on 
solar time in one solar hour, and therefore the term I>'.8.5G5 X 7^ 
is the correction to add to 7{,t() reduce it from a solar to a side- 
real interval. This term is identical with (/i — 1)7)^ as given by 

* The Midcrenl time at mean noon is equal to the tntf R.A. of the mean sun, or it 
is the R.A. of the mean sun referred to the truf equinox, ami therefore inToWes the 
nutation, so that its rate of increase is not, strictly, uniform. But it is sufBcientlj m 
for 24 hours to be so regarded in all practical computations. See Chapter XI. 

TIME. 61 

tlie preceding article, if T^ in the latter expression is expressed 
in seconds, since wc have 

-?^:?^ :- 0.0027391= /I -1 

"We may then write (56) in the following form, putting L = the 
west lon^tude of the given meridian, and T^J= T+ L: 

e=!r+r, + (/x-i)(r+i) (57) 

The term (ji — 1) {T + L) is given in the tables of the Amer- 
ican Ephemeris for converting "Mean into Sidereal Time,*' and 
may be found by entering the table with the argument T + A 
or by entering successively with the arguments T and L and 
adding the corrections found, observing to give the correction 
for the longitude the negative sign when the longitude is east. 
If no tables are at hand, the direct computation of this term will 
be more convenient under the form 9".8565 X ^^o- 

Example 1.— In Longitude 165° W. 1856 May 17, 4* A.M.; 
what is the sidereal time ? 

The Greenwich time is May 17, 3*; and the computation may 
be arranged as follows : 

Local Ast. Time T == 10» 0- 0*. 
At Gr. Noon May 17, Vo= 3 41 28 .32 
Correction of V^ for 3* ) <^q ;.- 

= 9*.8565 X 3 

e= 19 41 57.89 

Example 2.— Li Longitude 25° 17' E. 1856 March 13, about 
9* 30* P.M., an observation is noted by a Greenwich chronometer 
which gives 7* 51"* 12'.3 and is slow 3"* 13\4 ; what is the local 
sidereal time ? 

Gr. mean date, March 13, 7* 54* 25*.7 

Longitude, 1 41 8 E. 

T = 9~~35~33T 

March 13, ro=23 25 12 .26 

Tabular corr. for 7* 54- 25v7 = 1 17 .94 

e =~9 2 3T90 


Example 3.— In Longitude 7* 25" 12- E. 1856 March 13, 18* 15- 
47'.3 mean local astronomical time ; what is the sidereal time? 

T =13»15-47'.3 
F„=23 25 12.26 
Tabular corr. for 13* 15- 47v3 = + 2 10 .73 
Tab. corr. for long. — 7* 25- 12-. = ~ 1 13.14 

e = 12 41 57 .15 

51. To convert the apparent solar time at a given meridian into the 
sidereal time at that meridian. 

Find the mean time by Art. 48, and then the sidereal time by 
Art. 50. 

Or, more directly, to the given apparmt time add the true sun*s 
right ascension. For if in Fig. 1 we take D as the true sun, we 
have DQ ^= apparent solar time, VD = R. A. of true sun, and 
VQ, the sidereal time, is the sum of these two. 

The right ascension of the true sun is called in the Ephemeris 
the "sun's apparent right ascension,'* and is there given for each 
apparent noon. It is not a uniformly increasing quantity; but 
for many puq^oses it will be sufficiently accurate to consider the 
hourly increase given in the Ephemeris as constant for 24*, and 
to add to the app. K. A. of the Ephemeris the correction found 
by multiplying tlie hourly difterence by the number of hours in 
the Greenwich time. 

Example.— In Longitude 98° W. 1856 June 3, 4* 10« P.M. 
app. time ; what is the sidereal time ? 

(ir. app. date June 8, 10* 42- (^ 10*.7) Local app. t. — 4* 10- 0*. 

O's App. K. A. App. noon June 3 = 4 46 22 .04 
Hourly diff. = 10'.271 Corr. :^ 10\271 X 10.7 = 1 49 .9 

Sidereal time =8 58 11 .94 

52. To conrcrt the sidereal time at a given meridian into the mean 
time at that meridian. 

First ynethod. — AVhen the Greenwich mean time is also given, 
as is fro(picntly the case, we have (»nly to find V as in Art. 50 
by a<Ming to Vj, given in the Ephemeris the correction for the 
(ireenwich time taken from the table ''Mean into Sidereal 
Time," and then we have, by transposing cipiation (56), 

T :- e - V 

TIME. 63 

Example. — In Longitude 165° W., the Greenwich mean time 
being 1856 May 17, 3*, the local sidereal time 19* 41*^ 57*.89, 
what is the local mean time ? 

Fo = 3» 41- 28-.32 
Corr. for 3* r= + 29.57 

r = 3 41 57 .89 
e =z 19 41 57 .89 

6 — F= T = 16 0.00 
The longitude being 11* W., the local date is May 16. 

Second method. — Wlien the Greenwich mean time is not given, 
we can find T from (57), all the other quantities in that equation 
being known. Wo find 

or, in a more convenient form for use, 

r=e-F.-(i-l)(e-F. + i) (58) 

in which the term multiplied by 1 — — is the retardation of mean 

time on sidereal in the interval © — l^o + ^^y ^^^^^ '^^ given in the 
table "Sidereal into Mean Time." It is convenient to enter the 
table first with the argument © — V^ and then with the argu- 
ment //, and to subtract the two corrections from © — V^^ ob- 
serving that the correction for the longitude becomes additive 
if the longitude is east. 

Example.— In Longitude 165° W. 1856 May 16, the sidereal 
time is 19* 41"* 57'.89; what is the mean local time? 

e = 19* 41-» 57v89 
May 16, Y^ = 3 ^7_3K76 

e — T; =T6 ~T~26~13 

Table, "Sidereal into | Corr. for 16* 4- 26'.13 = — 2 38 .00 

Mean Time" I " " longitude 11* == _ 1 48 .13 

r="i6 oToo 

63. The following method of converting the sidereal into the 

mean time is preferred by some. In the last column of page III 

of the month in the American Naut. Aim. is given the **Mean 

Time of Sidereal 0*.*' This quantity, which we may denote by 

F', is the number of hours the mean sun is west of the vernal 

64 TIME. 

equinox, and is merely the difterence between 24* and the mean 
Bun*8 right ascension. The hour angle of the mean sun at any 
instant is then the hour angle of the vernal equinox increased 
hy the value of F' at that instant. To find this value of V% we 
first reduce the Almanac value to the given meridian by cor- 
recting it for the longitude by the table for converting sidereal 
into mean time; then reduce it to the given sidereal time © 
(which is the elapsed sidereal time since the transit of the vernal 
equinox over the given meridian) by further correcting it by the 
same table for this time 0. We then have the mean time 7^ by 
the formula 

It is necessary to observe, however, that if + V'' exceed 
24* it will increase our date by one day; and in that case V 
should be taken from the Ahnanac for a date one day less than 
the given date; that* is, we must in every case take that value 
which belongs to the Greenwich transit of the vernal equinox 
immediately j^rctu^dinf/ that over the given meridian. 

Example. — Same as in Art. 52. 

e = 19* 41- 57*.89 

May 15, TV = 20 23 3 .88 

Corr. for long. 11* W. ..:. — 1 48 .13 

Corr. for 19* 41" 58- = —3 13 .04 

T^='i0 ~0 0.00 

54. To find the hour angk of a star* at a given tunc at a given 

In Fig. 1, we have for the star at 0, DQ -- VQ — VD\ that 
is, if wc put 

the sidereal time, 

a . the right ascension of the star, 

t =--. the hour angle 

<< a. a 

then t -, e - - a (59) 

If a exceeds 0, this tormula will give a negative value of i 
which will express tlu* hour angle east of the meridian: in that 
case, if we increa.«<e bv 24* before subtractintc Gt, we shall find 

♦ W<? shall uxe "star," for brevity, to denote any cck'!<tiul hody. 


the value of t reckoned in the usual manner, west of the meri- 

According to this formula, then, we have first to convert the 
given time into the sidereal time, from which we then subtract 
the right ascension of the star, increasing the sidereal time by 
24* when necessary; the remainder is the required hour angle 
west of the meridian. 

In the case of the sun, however, the apparent time is at once 
the required hour angle, and we only have to apply to the given 
mean time the equation of time. 

Example.— In Longitude 165^ W. 1856 May 16, 16* 0« O' mean 
time, find the hour angles of the sun, the moon, Jupiter, and 
the star Fomalhaut. 

The Greenwich mean date is 1856 May 17, 3*, and the local 
sidereal time is (see Example 1, Art. 50) = 19* 41- 57'.89. 
For the Greenwich date we find from the Naut. Aim. the equa- 
tion of time Ey and the right ascensions a of the moon, Jupiter, 
and Fomalhaut, as below : 

T = 16* 0* 0* e = 19* 41* 57*.89 

— ^= + 3 49.85 y8a = 13 50 21.35 

O's ^ = 16 3 49 .85 

e = 19* 41- 57*.89 
Ql's a = 7 57.52 

Ql's t = 19 34 .37 Fomalh. ^ = 20 52 17 .71 

K the sidereal time had been given at first, we should have 
found the hour angle of the sun by subtracting its apparent right 
ascension as in the case of any other body. 

55. Given the hour angle of a star at a given meridian on a given 
day J to find the local mean time. 
By transposing the formula (59), we have 

e = < + a ' (60) 

so that, the right ascension of the star being given, we have only 
to add it to the given hour angle to obtain the local sidereal time, 
whence the mean time is found by Art. 52. When the sum t + a 
exceeds 24*, we must, of course, deduct 24*. If the body is the 
sun, however, the given hour angle is at once the apparent time, 
whence the mean time as before. But if the body is the moon 

Vol. L— 6 


t — 




O — 









66 TIME. 

or a planet, its right ascension can be found from the Ephemeris 
only when we know the Greenwich time. If then the Green- 
wich time is not given, we must find an approximate value of 
the local time by formula (60), using for a a value taken for a 
Greenwich time as nearly estimated as possible ; from this local 
time deduce a more exact value of the Greenwich time, with 
which a more exact value of a may be found ; and so repeating as 
often as maybe necessary to reach the required degree of precision. 

Example 1. — In Longitude 165® W. 1856 May 16, the hour angle 
of Fomalhaut is 20* 52-* 17'.71 ; what is the mean time ? 

t = 20* 52* 17'.71 
May 16, Fomalh. a = 22 49 40 .18 

e = 19 41 57 .89 
whence the mean time is found to be T= 16* 0* 0*. 

Example 2. — In Longitude 165° W. 1856 May 16, the moon's 
hour angle is 5* 51* 36*.54, and the Greenwich date is given May 
17, 3* ; what is the mean time ? 

t = 5» 51- 36*.54 
For May 17, 3», a = 13 50 21.35 

6 = 19 41 57.89 
" Mayl7, 3\ F= 3 41 57.89 

T = 16 .00 

Example 3. — In Longitude 30° E. 1856 August 10, the moon's 
hour angle is 4* 10* 53*.2; what is the mean time ? 

For a first approximation, we ob8er\'e that the moon passes the 
meridian on August 10 at about 7* mean time (Am. Eph. page 
IV of the month), and when it is west of the meridian 4* the 
mean time is about 4* later, or 11*, from which subtracting the 
longitude 2* we have, as a rough value of the Greenwich time 
Aug. 10, 9*. We then have 

t — 4» 11« 

For Aug. 10, 9*, 

a = 16 29 

6 — 20 40 

" Aug. 10, 9*, 

F— 9 18 

Ist approx. value 7= 11 22 

Hence the more exact Greenwich date is Aug. 10, 9* 22*; and 
with this we now repeat : 


t= 4*10*53-.2 
For Aug. 10, 9» 22- a = 16 29 26.8 

6 = 20 40 20.0 
»' « V= 9 18 8.1 

2d approx. valud T = 11 22 11 .9 

A third approximation, setting out from this value of T, gives 
us r= 11* 22- 12'.32. 

56. The mean time of the meridian passage not only of the 
moon but of each of the planets is given in the Ephemeris. 
This quantity is nothing more than the arc of the equator in- 
tercepted between the mean sun and the moon's or planet's 
declination circle. If we denote it by M^ we may regard J!f as 
the equation between mean time and the lunar or planetary time, 
these terms being used instead of "hour angle of the moon" or 
"hour angle of a planet," just as we use "solar time" to signify 
"hour angle of the sun." This quantity M is given in the Ephe- 
meris for the instant when the lunar or planetary time is 0*, and 
its variation in 1* of such time is also given in the adjacent 
column. If, then, when the moon's or a planet's hour angle at a 
given meridian = tj we take out from the Almanac the value of 
M for the corresponding Qreen^vich value of /, we shall find the 
mean time jTby simply adding Mio t; that is, 

T=t+M (61) 

This is, in fact, the direct solution of the problem of the pre- 
ceding article, and neither requires a previous knowledge of the 
Greenwich mean time nor introduces the sidereal time. But 
the Almanac values of M are not given to seconds ; and there- 
fore we can use (61) only for making our first approximation to 
Tj after which we proceed as in the last article. The Green- 
wich value of t with which we take out M is equal to < + i, 
denoting by L the longitude of the given meridian (to be taken 
with the negative sign when east), and the required value of M is 
the Almanac value increased by the hourly diff*. multiplied by 
{t + L) in hours. As the hourly diff. of M in the case of the moon 
is itself variable, we should use that value of it which corresponds 
to the middle of the interval t-\-L; that is, we should first correct 
the hourly diff. by the product of its hourly change into J (< + i) 
in hours. 


Example. — Same as Example 3, Art. 55. We have 

f 4. X = 2* 10- 53'.2 = 2M8 t = 4» 10- 53'.2 

At Gr. trans. Hour. Diff. =2".17 AtGr.traTis.Aug.lO,i/^=: 7 6 30 
Variation of H. D. in 1* 5- = .01 2-.18 X 2.18 = + 4 45 

Corrected Hourly Diff. = 2 .18 T= 11 22 8 .2 

which agrees within 4* with the true value. Taking it as a first 
approximation, and proceeding as in Art. 55, a second approxima- 
tion gives T = 11* 22^ 12M9. 


57. We have already had occasion to refer to the Ephemeris ; 
but we propose here to treat more particularly of its arrange- 
ment and use. 

The Astronomical Ephemeris expresses in numbers the actual 
state of the celestial sphere at given instants of time ; that is, 
it gives for such instants the numerical values of the co-ordi- 
nates of the principal celestial bodies, referred to circles whose 
positions are independent of the diurnal motion of the earth, 
as declination and right ascension, latitude and longitude; 
together with the elements of position of the circles of re- 
ference themselves. It also gives the eftects of changes of posi- 
tion of the observer upon the co-ordinates, or, rather, numbers 
from which such changes can be readily computed (namely, 
the parallax, which will be fully considered hereafter), the ap- 
parent angular magnitude of the sun, moon, and planets, and, 
in general, all those phenomena which depend on the time; that 
is, which may be regarded simply Rsfimr(io)}s of (he time. 

The Americav Ephemeris is composed of tsvo parts, the first 
computed for the meridian of Greenwich, in conformity with the 
British Nautical Almanac, especially for the use of navigators ; 
the second computed for the meridian of Washington for the 
convenience of American astronomers. The French Ephemeris, 
La Connaissance des Tanps, is computed for the meridian of Paris; 
the German, Berliner Astronomisches Jahrbuch, for the meridian 
of Berlin. All these works are published annually several years 
in advance. 

58. In what follows, we assume the Ephemeris to be computed 
for the Greenwich meridian, and, consequently, that it contains 
the right ascensions, declinations, equation of time, &c. for given 
equidistant instants of Greenwich time. 


Before we can find from it the values of any of these quanti- 
ties for a given local time, we must find the corresponding Green- 
wich time (Arts. 45, 46). When this time is exactly one of the 
instants for which the required quantity is put down in the Ephe- 
meris, nothing more is necessary than to transcribe the quantitj- 
as there put down. But when, as is mostly the case, the time 
falls between two of the times in the Ephemeris, we must obtain 
the required quantity by interpolation. To facilitate this inter- 
polation, the Ephemeris contains the rate of change, or difterence 
of each of the quantities in some unit of time. 

To use the difference columns with advantage, the Qreenwnch 
time should be expressed in that unit of time for which the 
difference is given : thus, when the difference is for one hour, 
our time must be expressed in hours and decimal parts of an 
hour ; when the difference is for one minute, the time should be 
expressed in minutes and decimal parts, &c. 

69. Simple interpolation, — In the greater number of cases in 
practice, it is sufficiently exact to obtain the required quantities 
by simple interpolation ; that is, by assuming that the differences 
of the quantities are proportional to the differences of the times, 
which is equivalent to assuming that the differences given in the 
Ephemeris are constant. This, however, is never the case; but 
the error arising from the assumption will be smaller the less 
the interval between the times in the Ephemeris ; hence, those 
quantities^ which vary most irregularly, as the moon's right 
ascension and declination, are given for every hour of Green- 
wich time ; others, as the moon's parallax and semidiameter, for 
every twelfth hour, or for noon and midnight ; others, as the 
sun's right ascension, &c., for each noon ; others, as the right 
ascensions and declinations of the fixed stars, for every tenth day 
of the year. Thus, for example, the greatest errors in the right 
ascensions and declinations found from the American Ephe- 
meris by simple interpolation are nearly as follows : — 

Error in R. A. 

Error In Decl. 






1 .5 






2 .4 



5 .4 


To illuBti^ate simple interpolation when the Greenwich time is 
given, we add the following 


For the Greenwich mean time 1856 March 30, 17* 11" 12*, 
find the following quantities from the American Ephemeris : 
thQ Equation of time, the Right Ascension, Declination, Hori- 
zontal Parallax, and Semidiameter of the Sun, the Moon, and 

1. The Equation of time,— The Gr. T. = March 30, 17* 11-2 = March 
30, 17M87. 

(Page II) E at mean noon = + 4- 27*.ll H. D. = — 0'.763 

Corr. for 17M9 = — 13.11 17.19 

^ = _|_ 4 U .00 — 13.11 

Note. — Observe to mark E always with the sign which denotes how it is to be 
applied to apparent time. If increafingy the II. D. (hourly difference) should hare 
the same sign as E; otherwise, the contrary sign. 

2. SurCs E, A. and Dec, 

(P. II.) a at 0* = 0* 36- 40'.78 H. D. + 9*.094 

Corr. for 17M87 = + 2 36.29 17.1^7 

a= 39 17.07 156.29 

^ at 0* = + 3° 67' 21".9 H. D. + 58'M5 

Corr. for 17M87 = + 16 39 .4 17.187 

^ = _^ 4 14 1 .3 999.4 

3. Moon's E, A, and Dec, 

a at 17* = 20* 18-' ^M Diff. 1- + 2'.4975 
Corr. for 11*2 = + 27 .97 11.2 

a= 20 18 37 .77 27.97 

d at 17* = — 25^ 3' 10".9 Diff. 1- + 8".275 

Corr. for 11-.2 = + 1 32 .7 11.2 

^ = — 25 1 38 .2 92.68 

4. Moon's Hor. Par, (= r) and Semid. (= S). 

n at 12* = 58' 44".l H. D. + 2".17 
Corr. for 5*.2 = + 11 .3 5^ 

r = 58 55 .4 11.28 


S at 12* = 16' 2".0 Diff. in 12* = + 7'M 

Corr. for 5*.2 = + 3 .1 

5=16 5 .1 

5. Jupiter's R, A. and Dec, 

a at 0* = 23* 29- 49'.95 H. D. + 2'.175 

Corr. for 17M87 = + 37.38 17.187 

a = 23 30 27.33 37.38 

^ at 0* = — 4« 22' 45".6 H. D. + 13".74 

Corr. for 17*.187 = + 3 56 .1 17.187 

^ = — 4 18 49 .5 236.1 

6. Jupiter's Hor, Par. and Semid. — At the bottom of page 231, we 
find for the nearest date March 31, without interpolation : 

It = 1".5 S = 15".7 

NoTK. — It maj be obserred that we mark hourlj differences of declination plut, 
when the body is moving northward, and minus when it is moving southward, 


In the above we have carried the computation to the utmost 
degree of precision ever necessary in simple interpolation. 

60. To find the right ascension and declination of the sun at the time 
of its transit over a given meridian^ and also the equation of time at 
the same instant. 

When the sun is on a meridian in west longitude, the Green- 
wich apparent time is precisely equal to the longitude, that is, 
the Gr. App. T. is after the noon of the same date with the local 
date, by a number of hours equal to the longitude. When the 
sun is on a meridian in east longitude, the Gr. App. T. is before 
the noon of the same date as the local date, by a number of 
hours equal to the longitude. Hence, to obtain the sun's right 
ascension and declination and the equation of time for apparent 
noon at any meridian, take these quantities from the Ephemeris 
(page I of the month) for Greenwich Apparent Noon of the 
same date as the local date, and apply a correction equal to the 
hourly difference multiplied by the number of hours in the lon- 
^tude, observing to add or subtract this correction, according as 
the numbers in the Ephemeris may indicate, for a time before or 
after noon. 


Example 1.— Longitude 167° 31' W. 1856 March 20, App. 
Noou, find O's R. A., O's Dec, and Eq. of T. 

Longitude = + 11* 10» 4* = + 11M7 

a at App. 0* = 0* 0* 20'.94 H. D. + O'.OOS 
Corr. for + 11M7 = + 1 41.62 + 11.17 

a= 2 2.56 + 101.62 

d at App. 0*= + 0° 2' 16".5 H. D. + 59".21 
Corr. for + 11M7 = + 11 1 .4 -f 11.17 

^ = -f- 13 17 .9 + 661.4 

E at App. 0* = + 7- 31*.57 H. D. — 0'.759 
Corr. for + 11M7 = — 8.48 + 11.17 

E = -{- 7 23 .09 — 848 

Example 2.— Longitude 167° 31' E. 1856 March 20, App. 
Noon, find O's R.A., 0*8 Dec, and Eq. of T. 

Longitude = -— 11* 10- 4*= — 11M7 

a at App. 0* = 0* 0- 20*.94 H. D. + 9*.098 
Corr. for — 11M7 = — 1 41 .62 -Z_^^'yL 

a = 23 58 39 .32 -^01j62 

d at App. 0* = + 0° 2' 16".5 H. D. + 59".21 
Corr. for — 11M7 = — 11 1 .4 — 11.17 

* d — 

8 44 .9 

— 661.4 

E at App. 0* — 

+ 7- 31*.67 

M. D. 0*.759 

Corr. for — 11*. 17 — 

+ 8.48 


J^ = 

+ 7 40 .05 

+ 8.48 

61. To find the mean local time of the moon's or a planefs transit 
over a givai meridian. 

This ift the flame as the problem of Art. 55, in the Bpecial case 
where the hour angle of the moon or planet at the given meri- 
dian is 0*. We can, however, obtain the required time directly 
from the Ephemeris, with sufficient accuracy for many purposes, 

* In this example the sun crosses the equator between the times of its transits 
OTer the local and the Greenwich meridians. The case must he noted, as it is a f^- 
quent occasion of error among navigators. The same case can occur on September 
22 or 28. 


by simple interpolation. On page IV of the month {Am. Ephem. 
and British NauU Aim.) we find the mean time of transit of the 
moon over the Greenwich meridian on each day. This mean 
time is nothing more than the hour angle of the mean sun at 
the instant, or the difference of the right ascensions of the moon 
and the mean sun ; and if this difference did not change, the 
mean local time of moon's transit would be the same for all 
meridians; but as the moon's right ascension increases more 
rapidly than the sun's, the moon is apparently retarded from 
transit to transit. The difference between two successive times 
of transit given in the Ephemeris is the retardation of the moon 
in passing over 24* of longitude, and the hourly difference given 
is the retardation in passing from the Greenwich meridian to 
the meridian 1* from that of Greenwich. Hence, to find the 
local time of the moon's transit on a given day, take the time of 
meridian passage from the Ephemeris for the same date (astro- 
nomical account) and apply a correction equal to the hourly 
difference multiplied by the longitude in hours; adding the 
correction when the longitude is west, subtracting it when east. 
The same method applies to planets whose mean times of transit 
are given in the Ephemeris as in the case of the moon. 

Example.— Longitude 130° 25' E. 1856 March 22 ; required 
local time of moon's transit. 

Gr. Merid. Passage March 22, 13*. 2«.7 H. D. + V^.m 
Corr. for Long. — 8*.7 = — 13.8 — 8J 

Local M. T. of transit = 12 48.9 — 13.8 

62. To find the moon's or a planefs right ascension, declination, 
^c.y at the time of transit over a given meridian. 

Find the local time of transit by the preceding article, deduce 
the Greenwich time, and take out the required quantities from 
the Ephemeris for this time. This is the usual nautical method, 
and is accurate enough even for the moon, as meridian observa- 
tions of the moon at sea are not susceptible of great precision. 
For greater precision, find the local time by Art. 55 for t = 0*, 
and thence the Greenwich time. See also Moon Culminations, 
Chapter VU. 

63. Interpolation by second differences. — The differences 
between the successive values of the quantities given in the 


Ephemeris as functions of the time, are called the first differ- 
mccs; the differences between these successive differences are 
called the second differences; the differences of the second differ- 
ences are called the third differences^ &c. In simple interpolation 
we assume the function to vary uniformly ; that is, we regard 
the first difference as constant, neglecting the second difference, 
which is, consequently, assumed to be zero. In interpolation 
by second differences we take into account the variation in the 
first difference, but we assume its variations to be constant; 
that is, we assume the second difterences to be constant and the 
third differences to be zero. 

AVTien the American Ephemeris is employed, we can take the 
second difterences into account in a very simple manner. In 
this work, the difterence given for a unit of time is always the 
difterence belonging to tlie instant of Greenwich time against 
which it stands, and it expresses, therefore, the rate at which 
the function is changing at that instant. This difterence, which 
we may here call the first difterence, varies with the Greenwich 
time, and (the second difterence being constant) it varies uni- 
formly, so that its value for any intermediate time may be found 
by simple inteq^olation, using the second difterences as first dif- 
ferences. Now, in computing a correction for a given interval 
of Greenwich time, we should employ the meany or average 
value, of the first difterence for the interval, and this mean 
value, when we regard the second differences as constant, is 
that which belongs to the middle of the interval. Hence, to 
take into account the second difterences, we have only to obser\'e 
the very simple rule — employ that (interpolated) value of the first 
difference which corresponds to the middle of the interval for ichich the 
correction is to be computed. 

Example. — For the Greenwich time 1856 March 2, 12* 29* 36% 
find the moon's declination. 

March 2. V2^{A) :^ — 27° 10'41".8 DiflF. 1- .^ -f 4".814 2d Diff. = + (r.l89 

Corr. for 29'».6 -f 2 23 .9 Corr. for2dd iff. -f .047 0.26 

<J= —27 8 17 .9 -f 4.861 -f 0.047 


+ 143.89 

Here the "diff. for 1-" increases 0'M89 in 1*; the half of the 
interval for which the correction is to be computed is 14" 48* = 


0*.25; we therefore find the value of the first difference at 12* 
14" 48% by adding to its value taken for 12* the quantity 0'M89 
X 0.25, and then proceed as in simple interpolation. This exam- 
ple suffices to illustrate the method in all cases where the first 
difference is given in the Ephemeris for the time against which 
it stands. In using the British Nautical Almanac and other 
works of the same kind, interpolation by second differences 
may be performed by the general interpolation formula here- 
after given. 

64. To find the Greenwich time corresponding to a given right ascen- 
sion of the moon on a given day. 

Let T' = the Greenwich time corresponding to the given right 

ascension a', 
T = the Greenwich hour preceding T' and corresponding to 

the right ascension a, 
Aa = the diff. of E. A. in 1* at the time T, 

then we have, approximately, 

rpr rp °' — * 

To correct for second differences, we have now only to find 

Aytt = diff. of E.A. in 1* for the middle instant 
of the interval T—T, 

and then we have, accurately, 

rpf rp * — * 


These formulse give T' — T in minutes of time. 

65. To find the distance of the moon from a given object at a given 
Greenwich time. 

In the American Ephemeris and the British Nautical Alma- 
nac, the "lunar distances** are given at every 3d hour of Green- 
wich time, together with the proportional logarithms of the differ- 
ences between the successive distances. 

The proportional logarithm of an angle expressed in hours, 
&c. is the logarithm of the quotient of 3* divided by the angle ; 
that of an angle expressed in degrees, &c. is the logarithm of 
the quotient of 3° divided by the angle. Thus, if A is the angle, 
in hours. 



P. L. -4 = log- = log3* —log^ 


or, if A is in degrees, 

P. L. ^ = log-=log3^ — log^ 


The angle is always supposed to be reduced to seconds ; so that, 
whether A is in seconds of time or of arc, we have 

P. L. ^ = log 10800 — log A 

Tables of such logarithms are given in works on Navigation. 

If now we wish to interpolate a value of a lunar distance for a 
time T-\- 1 which falls between the two times of the Ephemeris 
7" and T+ 3*, we are to compute the correction for the interval t 
and apply it to the distance given for the time T; and if we put 

J =tbe difference of the distances in the Ephemeris^ 
J' = the difference in the interval f, 

we shall have, by simple interpolation, 

J'= JX- 

or, by logarithms, 

log J' = log t + log J — log 3* 

or, supposing J, J', and i all reduced to seconds, 

log J' = log f — P. L. J (62) 

Subtracting both members of this from log 10800, we have 

P. L. J' = P.L. f + P. L. J (63) 

which is computed by the tables above mentioned. By (62), 

however, only the common logarithmic table is required. 

But the first difterences of the lunar distance cannot be assumed 

as constant when the intervals of time are as great as 3\ If 

we put 

P. L. J = § 

we observe that Q is variable, and the value given in the Ephe- 
meris is to be regarded as its value at the middle instant of the 
interval to which it belongs. If then 

Of = the value of Q for the middle of the interval t, 
^Q = the increase of Q in 3* (found from the successive values 
in the Ephemeris), 


we have 

«.=e-(!^').« (64) 

in which t is in hours and decimal parts. We find then, with 
regard to second differences, 

log J' = logt — Q' 

Example. — ^Pind the distance d of the moon's centre from the 
star Fomalhaut at the Greenwich time 1856 March 80, 18* 20»* 

Here T= 12*, e = 1* 20- 24* = 1*.34 : ^*'^ """ * ^ = 0.28 : and from the 
Ephemeris : 

March 30, 12* ((0 36^ 17' 53" Q, .2993 a C, + 0041 

J' — 40 28 — jOOn ,2S 

At 13* 20- 24* 4 = 35 37 25 ^, .2982 + .0011 

lege, 3.6834 

log J', 3.3852 

66. To find the Greenwich time corresponding to a given lunar dis- 
tance on a given day. 

We find in the Ephemeris for the given day the two distances 
between which the given one falls; and if J' = difference be- 
tween the first of these and the given one, J = difference of the 
distances in the Ephemeris, we find the interval ^, to be added to 
the preceding Greenwich time, by simple interpolation, from the 


t = 3*XT 


logf = log J' + P.L. J = log J'4- C (65) 

and, with regard to second differences, the true interval, <', by 

the formula 

log^ = logJ'+e' (66) 

where Q' has the value given in the preceding article. 

But to find C' by (64) we must first find an approximate value 
of t. To avoid this double computation, it is usual to find t by 
(65), and to give a correction to reduce it to <' in a small table 
which is computed as follows. We have fipom (64), (65), and (66) 


log f -log t=Q'-Q = -\^^^^^^yQ 

By the theory of logarithms, we have, J!f being the modulus 
ofthe common system, 

log a: = if [(a: — 1) — J (^ — 1)" + &C.] 
so that 

logf-logt = log'-=M^^'-\{'-f-') + kc:^ 
or, neglecting the square and higher powers of the small fraction 

This, substituted above, gives 

jlfX3* ^ 2.VX3* ^ 

by which a table is readily computed giving the value of V — i 
[or the correction of / found by (65)], li^'ith the arguments aQ and L 
Li this formula t and V — t are supposed to be expressed in hours; 
and to obtain t' — i in seconds we must multiply the second 
member by 3600 ; this will be effected if we multiply each of the 
factors t and 3* — < by 60, that is, reduce them each to minutes, 
so that if we substitute the value of M^= .434294 the formula 

^., = «^^-ZlO^(? (67) 

2.60676 ^ ^ ^ 

in which t is expressed in minutes, and V — tin seconds. 

Example. — 1856 March 30, the distance of the moon and 
Fomalhaut is 35° 37' 25" ; what is the Greenwich time ? 

March 30, 12* 0* 0- ((/)=36° 17' 53" Q= .2993 A§= + 41 
f= 1 20 3 6 d =35 37 25 log J' = 3.3852 

Ap. Gr. time =13 20 36 J' 40 28 \ogt =3.6845 

By (07)*,f'— f = —12 

True Gr. time = 13 20 24 

* Or from the ** Table showing (he correction required on account of the seeond 
differences of the moon's motion in finding the Greenwich time corresponding to % 
corrected lunar distance/' which is giyen in the American Ephemeris, and is also 
included in the Tables for Correcting Lunar Distances giTen in Vol. 11. of thla work. 




67. When the exact value of any quantity is required from the 
Ephemeris, recourse must be had to the general interpolation 
formulae which are demonstrated in analjiiical works. These 
enable us to determine intermediate values of a function from 
tabulated values corresponding to equidistant values of the 
variable on which they depend. In the Ephemeris the data are 
in most cases to be regarded as functions of the time considered 
as the variable or argument. 

Let T^ T-\' Wj 7^+ 2x0^ 7^+ 3t(7, &c., express equidistant values 
of the variable ; F^ F\ F'\ i^'", &c., corresponding values of 
the given function ; and let the difterences of the first, second, 
and following orders be formed, as expressed in the foUo^nng 
table : — 



let Diff. 

2d Diff. 

3d Diff. 

4th Diff. 

6t]i Diff. 

6th Diff. 




T-\- w 




















T + 4w7 













The differences are to be found by subtracting downwards^ that 
is, each number is subtracted from the number below it, and the 
proper algebraic sign must be prefixed. The difterences of any 
order are formed from those of the preceding order in the same 
manner as the first differences are formed from the given func- 
tions. The even differences (2d, 4th, &c.) fall in the same lines 
with the argument and function ; the odd differences (1st, 3d, &c.) 
between the lines. 

Now, denoting the value of the function corresponding to a 
value of the argument T+ mo by F^''\ we have, from algebra, 

jr(»)=JP+na+^ (^-^^ H^ ^^~^^ ^""^^ c+^ ^^-^^ ^^-^^ ^^-^^ d+kc, (68) 

1.2 1.2.3 

in which the coefficients are those of the n** power of a binomial. 



In this formula the interpolation sets out from the first of the 
given functions, and the differences used are the first of their 
respective orders. If n be taken successively equal to 0, 1, 2, 3, 
&c., we shall obtain the functions F^ F% -P", F^'% &c., and in- 
termediate values are found by using fractional values of n. We 
usually apply the formula only to interpolating between the 
function from which we set out and the next following one, in 
which case n is less than unity. To find the proper value of n 
in each case, let T+ i denote the value of the argument for which 
we wish to interpolate a value of the function : then 



n = — 


that is, n is the value of t reduced to a fraction of the interval w. 

Example. — Suppose the moon's right ascension had been 
given in the Ephemeris for every twelfth hour as follows : 

l«t. Diff. 

D *H R. A. 

1856 Blarch 6, 


21* 58" 28».89 



22 27 15.43 


22 65 25.50 



23 23 8.89 


23 50 15.63 



17 9.83 

2d DHL 

3d Diff. 

4th Diff. 1 

— 86».97 

H- 4'. 79 



.+ 1'.74 





-f 28* 47'.04 

28 10.07 

32.18 ' -I-K74 

27 37.89 6.53 — 0».66 

27 12.24 

26 54.20 

Required the moon's right ascension for March 5, 6*. 

6* 1 
Here 7"= March 5, 0*. / = 6\ w = 12\ n = --— = -; and if we 

denote the coeflicients of a, 6, c, rf, e in (68) by -4, -S, C, -D, U, 

we have 

J^ = 21* 58- 28'.39 

a = + 28- 47'.04, A=n = J, ^a = + 14 23.52 

36.97, 5=^.^^ = — J, Bb = + 

b=z — 

c = + 

4.79, C=^.^^— ?== + tV Cc= + 

d = + 1.74, D=C. 


= -T^B. ■»'* = - 

0.66, E=J).^^—^ = + ,ig,Ee= — 





J'» R A. 1856 March 5, 6» J^'^' = 22 12 66 .74 



which agrees precisely with the value given in the American 

68. The formula (68) may also be written as follows : 


d -| / e + &c. 

2 \ 8 \ 4 \ 6 

Thus, in the preceding example, we should have 


n — 4 

n— 3 


n — 1 


- /u X - 0'.66 = 

- I (+ 1-.74 + 0-.46) = 

- ^ (+ 4'.79 — 1'.38) = 

- 1.38 

— 1.71 

n = 


— I (— 36..97 — 1'.71) = + 9 .67 
i (+ 28«47*.04 + 9*.67) = + 14* 28'.35 

and adding this last quantity, 14"» 28'.35, to 21* 58*» 28'.39, we 
obtain the same value as before, or 22* 12"* 66*. 74. 

69. A more convenient formula, for most purposes, may be 
deduced from (68), if we use not only values of the functions 
following that from which we set out, but also preceding values ; 
that is, also values corresponding to the arguments T — Wj 
T — 2m?, &c. We then form a table according to the following 
schedule : 








6th Diff. 

T 3t£? 






T— w 













r+ w 












Vol. I.— 6 


According to the formula (68), if we set out from the function 
F^ we employ the differences denoted in this table by a', 6', c", 
&c., and hence for the argument T + nwvfQ find the value of 
F^*"^ by the formula 

jrc)^ir+na-+^L(!!z:l) y+ ^ ^"^^^ ^^^) n (n-1) (n^2) (n-3) ^ ^^^ 
^ ^ 1.2 1.2.8 

But we have 

d!'= d' ^e* = d +€f +€"+/' = d + 2ef+f 
&c. &c. 

in which b\ c", &c. are expressed in terms of the differences 
that lie on each side of a horizontal line drawn in the table 
immediately under the function from which we set out. These 
values substituted in the formula give 

^ ^ 1.2 1.2.3 

(n + l)(n)(n-l)(n-2)^ ^^ (69) 

^ ^ ^ 

in which the law of the coefficients is that one new factor is 
introduced into the numerator alternately after and before the 
other factors, observing always that the factors decrease by unity 
from left to right. The new factor in the denominator, as in the 
original formula (G8), denotes the order of difference. 

The inteq>olation by this formula is rendered somewhat more 
accurate by using, instead of the last difference, the mean of the 
two values that lie nearest the horizontal line drawn under the 
middle function : thus, if we stop at the fourth difference, we 
use a mean between d and r/' instead of d. We thus take into 
account a part of the term involving the fifth difterenee. 

Example. — Find the moon's right ascension for 185G March 5, 
6*, cnii»l()ying the values given in the Ephemeris for every 
twelfth hour. This is the same as the example un<ler Art. 67, 
where it is worked by the primitive fonnula (<)8). But here we 
take from the Ei>hemeris three values preceding \\\i\i for March 5, 
0*, and three values /oWo^nW/ it, and form our table as follows: 



1866 March 8, 12* 








D'gR. A. 

20* 28"» 17'.88 

20 58 57.08 

21 29 2.01 
21 58 28.39 

l8t DIff. 

2d Diff. 

3d Diff. 

4th Diff. 

-1- 80* 89'.20 

— 34'.27 

80 4.93 

— 4'.28 


+ 3'. 49 

29 26.88 

— 0.79 



5th Diff. 

— O-.SS 

6, 12 

22 27 15.43 

22 55 25.50 

23 28 8.89 

28 47.04 




28 10.07 


+ 4.79 

27 87.89 

— 0.74 

Drawing a horizontal line under the function from which we 
Bet out, the differences required in the formula (69) stand next 
to this line, alternately below and above it. 

a' = + 28- 47'.04, 

b = — 

</ = + 

d = + 

e = — 

39 .34, 

A = 
B = A 


2.37, C = B 
3.16, D=C 

0.74, E=D 

n + 1 

n— 2 

n + 2 

F — 21» 58- 28'.39 


Aa' = + 14 23 .52 


£b =+ 4.92 


CV — 0.15 

= + lis. I>d= + 

= + r* 


E^ = — 



D's E. A. 1856 March 6, 6* = jP«> = 22 12 56 .74 

69*. K in (69) we substitute the values 

a' = a^-\-b 
(/ = c, + d 
we find 

J.- = F+na,+ (i^±i^6 +(» + D W (n - 1) 
^ '^ 1.2 1.2.3 ' 

(n + 2) (n + 1) (n) (n - 1) 

+ 1727374 ^ + ^'- 


in which the law of the coefficients is that one new factor is 
introduced into the numerator alternately before (aid after the 
other factors, obsen'ing still that the factors decrease by unity 
from left to right. The differences employed are those which lie 
on each side of the horizontal line drawn immediately above 
the function from which we set out. 


If in the preceding formulae we employ a negative value of 
n less than unity, we shall obtain a value of the function between 
F and F^y and in that case (70) is more convergent than (69). In 
general, if we set out from that function which is nearest to the 
required one, we shall always have values of n numerically less 
than J, and we should prefer (69) for values of n between and 
+ J, and (70) for values of n between and — J, 

70. If we take the mean of the two formulse (69) and (70), 
and denote the means of the odd difterences that lie above and 
below the horizontal lines of the table, by letters without ac- 
cents, that is, if we put 

a = }(«, + a'), c=\{c^^€f) &c. 
we have 

F...=i.+„a+^ b+ (!Hd)00(-zl),^ («±1X"!)(ILZL) d + &c. (71) 
^ ^2 2.3 2.3.4 ^ ^ ^ 

The quantities a, c, &c. may be inserted in the table, and will 
thus complete the row of difterences standing in the same line 
with the function from which we set out. 

The law of the coefticients in (71) is that the coefticient of any 
odd difterence is obtained from that of tlie preceding odd dif- 
ference by introducing two factors, one at the beginning and 
the other at the end of the line of factors, observing as before 
that these factors are respectively greater and less by unity than 
those next to which they are placed; and the coefticients of the 
even difterences are obtained from the next preceding even 
difterences in the same manner. The factors in the denominator 
follow the same law as in the other formuljs. 

Example. — Find the moon's right ascension for 1856 March 5, 
6\ from the values given in the Ephcmeris for noon and mid- 

The table will be as below: 



Mar. 3, 12» 
" 4, 
« 4, 12 

" 6, 

" 5, 12 

" 6, 

" 6, 12 

D'8 R. A. 

20*28- 17-.88 

20 58 57 .08 

21 29 2 .01 

21 58 28.39 

22 27 15 .43 

22 55 25 .50 

23 23 3 .39 

1st Diff. 

2(1 Diff. 

8d Diff. 

4th Diff. 

+ 30«29'.20 

— 34-.27 

30 4 .93 




29 26.38 


[+29 6 .71] 


[+0 .79] 

+ 3.16 

28 47.04 

+ 2.37 



28 10.07 


+ 4.79 

27 37.89 

5th Diff. 

— 0'.33 

[— .54] 


Drawing two lines, one above and the other below the func- 
tion from wliich we set out, and then tilling the blanks by the 
means of the odd differences above and below these lines (which 
means are here inserted in brackets), we have presented in the 
same line all the differences required in the formula (71) ; and 
we then have 

b = — 

c= + 

d = + 

F— 21»58"28'.39 

» 6'.71, A— n 


Aa— + 14 33 .36 

39 .34, 5 — ^' 
' 2 

+ h 

Bh— 4 .92 

n«— 1 
0.79, C= A.—^=-ij,, 

3.16,2)= 5."^ = --,^^, 

Cc= — 

Dd = 

e=— 0.54, JE?=a. 



= + 2!ff, Ee=^ 




jp^oi) = 22 12 56 .75 

agreeing within O'.Ol with the value found in the preceding 
article. Hansen has given a table for facilitating the use of this 
formula. (See his Tables de la Lune). 

71. Another form, considered by Bessel as more accurate than 
any of the preceding, is found by employing the odd differences 
that fall next below the horizontal line drawn below the function 
from which we set out, and the means of the even differences 
that fall next above and next below this line. Thus, if we put 

b,= i(b + V), d, = \{d-\- d'), Ac. 



and combine these with the expressions 

i<f = i(^l/ ^b), i e' = 1( d'— rf), &c. 

we deduce 

b = 1^ — i (f, d = d^ — } e', &c. 


which substituted in (69) give 

^ ^ 1.2 "^ 1.2.3 ^ 

. (n+l)n(n-l)(n~2)(n-}) ^. , 

+ +*''• 



To facilitate the application of this formula, draw a horizontal 
line under the function from which the interpolation sets out, 
and another over the next following function ; these lines will 
embrace the odd differences a', c', &c. If we then insert in the 
blank spaces between these lines the means of the even differ- 
ences that fall above and below them, we shall have presented 
in a row all the diftercnces to be employed in the formula. 

Example. — ^Find the right ascension of the moon's second 
limb at the instant of its transit over the meridian whose longi- 
tude is 4* 42"* 19* west from Greenwich, on May 15, 1851. 

The right ascensions of the moon's bright limb at tlie instant 
of its upper and lower transits over the Greenwich meridian, are 
given in the Ephemeris, under the head of '* Moon Culminations." 
The argument in this case is the longitude, and the intervals of 
the ar*rument are 12*. The value for any meridian is therefore 
to be obtained by interpolation, taking for n the quotient obtained 
by dividing the given longitude (in hours) by 12*. 

AVe take from the British Nautical Almanac the following 
values : 

16* 12- 31>'.a4 
15 41 3.41 
IG 9 30.8<» 

lit THir. 



4th Diff. 

6th Diir. 

May 14. U. C. 
•* 1.'. L. C. 

" l',. u. c. 

J 28'" 




-f 12«.ll 
-f 9.49 

— 2'.62 

— K58 

10 38 !!.•>. 86 
17 7 17.12 
17 86 8.22 


45 .97 

-f- 6.29 
— 0.16 

- 4 .20 


-f 0».33 

" l(i, L. C. 
»* 10. u. c. 
- 17. L. C. 


61 .20 

— 6.46 

— 1.26 



For interpolation by formula (72) we draw a horizontal line 
below the function from which we set out, and one above the 
next following function. These lines enclose the odd difterences 
regularly occurring in the table. Inserting in the blanks in the 
columns of even diiFerences the means of the numbers above and 
below, all the differences to be employed in the formula stand in 
the same line, namely : 

a' = + 1725'.97, b^= + 7'.B9, c' = — 4'.20, d^ = — l'A2, e' = + 0*.33 

As n is here not a simple fraction, the computation will be 
most conveniently performed by logarithms, as follows : 

4» 42- 19* — 16939* 

log 4.2288878 

12* =43200 log 4.6354837 

log A — \ogn — 9.5934041 



9.59340 9.5934 1 9.5934 


n — 1 — 







— 0.10789 



n — 2 — 




n + l = 

+ 1.3921 




(^) 9.69^97 

(J) 9.2218 

(^'5) 8.6198 



(C) 7.6320 

(D) 8.3470 




(6o) 0.86864 



(e') 9.5185 



Aa' — 11- 16-.764 

Bb^ — .879 

Of =— 0.018 

Dd^ = .032 

E^ = 0.000 

Increase of E. A. — 11 15.835 

E. A. Greenwich Cn 
E. A. on given merit 

ilm. — 16» 9- 39-.890 

iian — 16* 

20- 55'.725 

The use of Bessel's formula of interpolation is facilitated by a 
table in which the values of the coefficients above denoted by 
Ay jB, Oy D, &c., and also their logaritlmis, are given with the 
argument 71. 

72. Interpolation into the middle. — ^When a value of the function 
is sought corresponding to a value of the argument which is a 


mean between two values for which the function is ^ven, that 
is, when » = |, we have by (72), since n — | = 0, 

or, since F+ia' = i {F+ F'), 

F^ = HF+ F') - i [6o - ^% [i„ - 2\ (/o - &c.)]] (73) 

which is known as the formula for interpolating into the middle. 

When the third differences are constant, rf^,, /q, &c. are zero, 
and the rule for interpolating into the middle between two func- 
tions is simply : From the mean of the two functions subtract one- 
eighth the mean of the second differences lohich stand against the func- 
tions. Interpolation by this rule is correct to third differences 

The formula (73) is especially convenient in computing tables. 
Values of the function to be tabulated are directly computed for 
values of the argument difiering by 2'"u?; then interpolating a 
value into the middle between each two of these, the arguments 
now difier by 2"*~*m? ; again interpolating into the middle between 
each two of the resulting series, we obtain a series with argu- 
ments differing by 2"*"*«/? ; and so on, until the interval of the 
argument is reduced to 2'*~"'ir or w. 

Example. — Find the moon's right ascension for 1856 March 
5, 6*, from the values of the Ephemeris for noon and midnight. 

This is the same as the example of Art. 69 ; but, as 6* is the 
middle instant between noon and midnight, the result will be 
obtained by the formula (73) in the following simple manner. 
We have from the table in Art. 69 

b^ = - 38'.16 ^(J^+ r) = 22» 12« 51'.91 
(/„ = + 2'.79, — -j»5 <^o = — ^J^ 38.68 X i = +4.83 

— 38.08 F'^^=22 12 56.74 

73. In case we have to interpolate between the last two values 
of a given series, we may consider the series in inverse order, 
tlie arguments being T, T—w, T—2w, &c., 7* being the last 
argument. The signs of the odd differences will then be changed, 
and, taking the last differences in the several columns as a, 6, c, d^ 
&€., the interpolation will be effected by (68). 


74. The interpolation formvlos arranged acco^'ding to the powers of 
the fractional part of the argument. 

When several values of the fuuetion are to be inserted between 
two of the given series, it is often convenient to employ the 
formula arranged according to the powers of n. Performing the 
multiplications of the factors indicated in (68), and arranging the 
terms, we obtain 

l?^*> = i?'+n(a — i^ + ^c — id + J« — Ac.) 
+ j^ (6 -c + H (f - I 6 + &c.) 

-4 (e — kc") 

^^ ^ 

+ &C (74) 

where the differences are obtained according to the schedule in 
Art. 67. 

Transforming (71) in the same manner, we have 

2?^-) = 7?^+ n (a - J c + 5I5 e — &c.) 

H (c — i e + &c.) 

^1.2.3^ ^ ^ 

H (d — &c.) 

^^ ^ 

H (e — &c.) 

^^ ^ 

+ &e (75) 

where the differences a, c, e, are the mean interpolated odd dif- 
ferences in the line of the function F of the schedule Art. 69. 

75. Derivatives of a tabulated function. — ^When the analytical ex- 
pression of a function is given, its derivatives may be directly 
found by successive differentiation ; but when this expression is 
not known, or when it is very complicated, we may obtain values 
of the derivatives, for particular values of the variable, from the 
tabulated values of the functions by means of their differences. 

Denoting the argument hy T + nWy its corresponding function 


by / ( 2' + w?(?), the successive derivatives of this function cor- 
responding to the same value of the argument will be denoted 
by f\T+nw\ f'\T+mo\ f'\T+ nw\ &c., and /(T), 
f\T)^ f'\T\ &c., will denote the values of the function and 
its derivatives corresponding to the argument T, or when n = 0. 
Hence, if we regard nw as the variable, we shall have, by Mao- 
laurin's Theorem, 

/(T + nw) = f{T) +fXT) nw +/"(r)^ + &c. 

Comparing the coefficients of the several powers of n in this 
formula with those in (74), we have 

/'(T) = — (a — } 6 + i c — M + i « — &c.) 

f'(T)=^(d — 2e + &c.) 


&c. &c (76) 

the differences being taken as in Art. 67. 

Still more convenient expressions are found by comparing 
Maclaurin's Theorem with (76) ; namely : 

fXT)=~ (a-Jo+,'o6-&c.) 

/"(r)=-l(6-yjd + &c.) 
/"'(r)=~ (c_tc + &c.) 



&c. &c. (77) 

the differenecH being found according to the schedule in Art 69, 
and the odd differences, a, c, e, &c., being interpolated means. 


The preceding formulae determine the derivatives for the value 
T of the argument. To find them for any other value, we have, 
by differentiating Maclaurin's Formula with reference to nw, 

/'(T + nw) =f\T) +f'iT) . nw + if'^T) . nW + &c. (78) 

in which we may substitute the values o{ f{T)jf'{T)y &c. from 
(76) or (77). 

In like manner, by successive diflerentiations of (78) we ob- 

/" iT+nw) =/" (T) +/'" (T). nw + if^ (T). n»M?' + &c. 
/'" (T + nw) ==/'" (T) +f' (T). nw + &c. 
&c. &c. 

76. An immediate application of (76) or (77) is the compu- 
tation of the differences in a unit of time of the functions in the 
Ephemeris ; for this difterence is nothing more than the first 
derivative, denoted above by the symbol /'. 

Example. — Find the difference of the moon's right ascension 
in one minute for 1856 March 5, 0*. 

We have in Art. 70, for T = March 5, 0*, a = 29" 6'.71, 
(? = + O-jg, € = — 0'.54, and m; = 12* = 720^ Hence, by the 
first equation of (77), 

f(T) = ^iji (29* 6'.71 — 0-.13 — 0'.02) = 2'.4258 

On interpolation, consult also Encke in the Jahrbuch for 1830 
and 1837. 


77. The Nautical Almanac gives the position of only a small 
number of stars. The positions of others are to be found in 
the Catalogues of stars. These are lists of stars arranged in 
the order of their right ascensions, with the data from which 
their apparent right ascensions and declinations may be ob- 
tained for any given date. 

The right ascension and declination of the so-called fixed 
stars are, in fact, ever changing: 1st, by precession, nutation, 
and aberration (hereafter to be specially treated of), which are 
not changes in the absolute position of the stars, but are either 
changes in the circles to which the stars are referred by sphe- 
rical co-ordinates (precession and nutation), or apparent changes 
arising from the observer's motion (aberration); 2d, by the 


proper motion of the stars themselves, which is a real change of 
the star's absolute position. 

In the catalogues, the stars are referred to a mean equator 
and a mean equinox at some assumed epoch. Tlie place of a 
star so referred at any time is called its mean place at that time ; 
that of a star referred to the true equator and true equinox, 
its true place ; that in which the star appears to the observer in 
motion, its apparent place. The mean place at any time will be 
found from that of the catalogue simply by applying the preces- 
sion and the proper motion for the interval of time from the 
epoch of the catalogue. The true place \\\\\ then be found by 
correcting the mean place for nutation ; and finally the appa- 
rent place will be found by correcting the true place for aber- 

To facilitate the application of these corrections, Bessbl pro- 
posed the following very simple arrangement. He showed 
that if 

a^,, d^ = the star's mean right asc. and dec. at the beginning of the 
a,d= the apparent right asc. and dec. at a time t of that year, 
T = the time from the beginning of the year expressed in decimal 
parts of a year, 
fi^ y! z=i the annual proper motion of the star in right asc. and dec. 


a z= a^^ Tfi + Aa + Bh -^ Cc + Bd -^ E \ 

d = d^+rii'+Aa' + Bb' + Cd + Dd! ] (™) 

in which a, 6, c, rf, a', 6', c', d' are functions of the star's right 
ascension and declination, and may, therefore, be computed for 
each star and given \\\\\i it in the catalogue ; -4, By C, /), E 
are functions of the sun's longitude, the moon's longitude, the 
longitude of the moon's ascending node, and the obliquity of the 
ecliptic, all of which depend on the time, so that -4, -B, C, D, E 
may be regarded 8imi)ly as functions of the time, and given in 
the Nautical Almanac for the given year and day; £ is a 
very small correction, usually neglected, as it can never ex- 
ceed 0".05. 

If the catalogue does not give the constants a, 6, c^ dy a', ft', c', 
d'y they may be computed, for the year 1850, by the following 
formulae (see Chap. XL p. 648): 


a = 46".077 + 20".056 sin a tan d a' = 20".056 cos a 

b = cos a tan d }/ = — sin a 

c = cos a sec a d = tan c cos ^ ~ sin a sin d 

rf =: sin a sec d d! = cos a sin d 

in which e = obliquity of the ecliptic. Or we may resort to 
what are usually called the independent constantSy and dispense 
with the «, i, e, rf, a', 6', (?', d' altogether, proceeding then by 
the formula 

« = ««+ ^M +/ +^ sin((y + a)tan^ + ^sin(^+a)8ec^l 

J == ^0 + '/*' +^ ^^8 ^ + ^ cos (G -j- a) 4" ^ cos (^+ a) sin ^ J ^ ^ 

the independent constants /, g^ G, A, If, i being given in the 
Epheraeris, together with the value of r for the given date, 
expressed decimally. 

It should be observed that the constants a, 6, c, rf, a', 6', c', d' 
are not absolutely constant, since they depend on the right 
ascension and declination, which are slowly changing : unless, 
therefore, the catalogue which contains them gives also their 
variations, or unless the time to which we wish to reduce is not 
very remote from the epoch of the catalogue, it may be prefer- 
able to use the independent constants. 

In forming the products Aa^ Bby &c., attention must of course 
be paid to the algebraic signs of the factors. The signs of J., i5, 
C, D are, in the Ephemerides, prefixed to their logarithms ; and 
the signs of a, 6, c, &c. are in some catalogues (as that of the 
British Association) also prefixed to their logarithms; but I 
shall here, as elsewhere in this work, mark only the logarithms 
of negative factors, prefixing to them the letter n. 

It should be remarked, also, that the B. A. C* gives the 

* B. A. C. — British Association Catalogue^ containing 8377 stars, distributed in aU 
parts of the heavens ; a very usefUl work, but not of the highest degree of precision. 
The Greenwich Catalogues, published from time to time, are more reliable, though 
less comprehensive. For the places of certain fundamental stars, see Bessel's 
TabuUe Regiomontanx and its continuation by Wolfers and Zech. 

Lalande*s Jlittoire Cileste contains nearly 50,000 stars, most of which are em- 
braced in a catalogue published by the British Association, reduced, under the 
direction of F. Baily, from the original work of Lalande. The Konigsberg Observa- 
tions embrace the series known as Bessel's Zones, the most extensive series of 
observations of small stars yet published. The original observations are given with 
dat« for their reduction, but an important part of them is given in Weis9b*s Posi- 
tiones Biediae Stellarum fixarum in Zonis Regiomontanis a Besselio inter — 15*^ et -{-Ib^ 
deelin. ohservat., containing nearly 32,000 stars. 

See also Struve*s Catal. generalis, and the catalogues of Aboelandeb, Rumker, 



north polar distance instead of the declination, or r^= 90*^ — 8^\ 
and, since ;r decreases when d increases, the corrections change 
their sign. This has been provided for by changing the signs of 
/i', rt', 6', c'j d' in the catalogue itself. Moreover, in this cata- 
logue, «, 6, a', b' denote Bessel's c, rf, c', d\ and tice versa ; and 
to correspond with this, the -4, B^ C, D of the British Almanac 
denote Bessel's C, -D, -4, ^. The same inversion also exists in 
the American Ephemeris prior to the year 1865, but in the volume 
for 1865 the original notation is restored. 

Example. — Find the apparent right ascension and declination 
of a Tauri for June 15, 1865, from Argelander's Catalogue. 
This star is Argel. 108 ; whence we take for 

Jan. 1, 1830. Mean R. A. = 4^ 2G« 10'.43 

Ann. prcc. = + S-. 428 1 ^^^ 35 ^^ 

Prop, motion =: -|- . 005 i 

= 4-2 0.15 5 

Jan. 1, 1805,0^=4 28 10.585 

Mean DecL = -f IS* y SS^.O 

— 0.17/ "^ 

4 30.55 

«t, = 4- 16 14 6 .55 

We next take the logarithms 

from the Catal. logs, a 0.5352 

from Am. Ephcm. ^ j^^ ^ ^ -g.. 
for June 15, ISGo, > 

from the Catal. logs, a' 0.8034 

h 7.8704 
B 0.0437 

b- n0.0607 

c 8.4320 

f* 0.2010 

d 8.S058 
D fil.3089 

^ 9.0378 

logs. Aa 0.3220 lib 38231 Cc n8.r,4.34 Dd »i0.1147 

logs. Aa' 0.C811 Bb' n0.9O44 Cc fiO.4144 /;<f ttO.3467 


Corr. of a^ yla = -i- 2M03. /?A = -f (:'.007, Cc ^ - 0».044, />J = — l'.S02 
Corr. of <V ^«' = -r •* "-80, i?6' = — 8 ".02, CV = — 0".2G, Dd ^ ^ 2".22 

We have also fi-om the catalogue /i ~ + 0*.005, //' — — O'MT. 
The fraction of a year for June 15, 18G5, is r — 0.4G ; and hence 

Jan. 1, 1865. a^ = 4* 28- 10».585 

Sum of corr. of a^ = + 0.824 

r/i = -L .(K)2 

June 15, 18»;5 a = 4~28 11 .411 

\ r^ -f 16° 14* 6".55 

Sum of corr. of t\ 



U -= 

- 5 .70 

— .08 

16 14 .77 

78. ^\lien the greatest precision is required, we should con- 
sider the change in the Htar s i»laoe even in a fraction of a day, 
and therefore aLso the change while the star is pa.«sing from ono 
meridian to another; also the set-ular variation and the changes 

PiAZZi. Santixi ; and the published ohitervationii of the principal obserratories. 8m 
also a list of catalogues in the introduction to the B. A. C. 


in the precession and in the logarithms of the constants. Fur- 
ther, it is to be observed that the annual precession of the cata- 
logues is for a mean year of 865"^ 5*.8. But for a fuller consider- 
ation of this subject see Chapter XI. 



79. The apparent positions of those heavenly bodies which are 
within measurable distances from the earth are diiFerent for ob- 
servers on different pai-ts of the earth's surface, and, therefore, 
before we can compare observations taken in diiFerent places we 
must have some knowledge of the form and dimensions of the 
earth. I must refer the reader to geodctical works for the 
methods by which the exact dimensions of the earth have been 
obtained, and shall here assume such of the results as I shall 
have occasion hereafter to apply. 

The figure of the earth is very nearly that of an oblate spheroid^ 
that is, an ellipsoid generated by the revolution of an ellipse 
about its minor axis. The section made by a plane through the 
earth's axis is nearly an ellipse, of which the major axis is the 
equatorial and the minor axis the polar diameter of the earth. 
Accurate geodctical measurements have shown that there are 
small deviations from the regular ellipsoid ; but it is sufficient 
for the puq^oses of astronomy to assume all the meridians to be 
ellipses with the mean dimensions deduced from all the measures 
made in various parts of the earth. 

80. Let EPQP\ Fig. 11, be one of the elliptical meridians of 
the earth, EQ the diameter of the equator, PP' the polar 
diameter, or axis of the earth, C the centre, F a focus of the 
ellipse. Let 

a = the semi-major axis, or equatorial radius, = CE, 
b = the semi-minor axis, or polar radius, = CP^ 
c = the compression of the earth, 
e = the eccentricity of the meridian. 


Br the romprcssitm is meant the difference of the equatorial 

and p»olar radii expressed in parts 
7i^ iL of the equatorial radius as unity, or 

a — 6 , b 
a a 

The (cccntrintjf of the meridian is 

r the distance of either focus from 

the centre, also expressed in parts 

of the equatorial radius, or, in 

Fig. 11, 

e ^ 


But, since PF= CE, we have. 

tbat is, 


CF* PP — PC^ PC* 





e» = l — - = 1 — (1— f)» 

e = I ' 2c — c* 


Bv a combination of all the most reliable measures, Bessel 
de^luced the most probable form of the spheroid, or that which 
luo-t nearly represents all the obse^^•ations that have been made 
in different parts of the world. He found* 

a ~ ^ "" 299.1528 


c = 

wh^mce, by ^81), 


t = .0816967 

lo^' /> = 8.912205 

log I (1 — ft ^ = 9.9985458 

• .-If/fr/ii'/nii^^A* Xarftnehf^n. No. 4.V. See «1j«o Encke's Tables of the dimensioBi 
of the terreitrial •pheroi<l id the Jnhrbuck for l8o2. 


The absolute lengths of the semi-axes, according to Bessel, are, 

a = 6377397.15 metres = 6974532.34 yds. = 3962.802 miles 
h = 6356078.96 *' = 6951218.06 " = 3949.555 »* 

81. To find the reduction of the latitude for the compression of the 

Let j4, Fig. 11, be a point on the surface of the earth; AT the 
tangent to the meridian at that point ; A 0, perpendicular to A Ty 
the normal to the earth's surface at A. A plane touching the 
earth's surface at A is the plane of the horizon at that point 
(Art. 3), and therefore AOy which is perpendicular to that plane, 
represents the vertical line of the observer at A. This vertical 
line does not coincide with the radius, except at the equator and 
the poles. If we produce CE^ OA^ and GA to meet the celestial 
sphere in E\ Z^ and Z* respectively, the angle ZO'E' is the 
declination of the zenith, or (Art. 7) the geographical latitude^ and 
Zis the geographical zenith ; the angle Z'CE' is the declination 
of the geocentric zenith Z'^ and is called the geocentric or reduced 
latitude; and ZAZ' = CAO is called the reduction of the latitude. 
It is evident that the geocentric is always less than the geogra- 
phical latitude. 

Kow, if we take the axes of the ellipse as the axes of co-ordi- 
nates, the centre being the origin, and denote by x the abscissa, 
and by y the ordinate of any point of the curve, by a and b the 
semi-major and semi-minor axes respectively, the equation of 
the ellipse is 

= 1 

If we put 

f> = the geographical latitude, 
fp' == the geocentric ** 

we have, since f is the angle which the normal makes with the 
axis of abscissae, 

tan f> = 

and from the triangle A CBy 

tan f ' = — 

Vol. L— 7 


Differentiating the equation of the ellipse, we have 

_y__ ^ dx 

X a' dy 


tan ^ = — tan f = (1 — e*) tan ^ (82) 

which determines the relation between f and f '. 

To find the difference y — f ', or the reduction of the latitude, 
we have recourse to the general development in series of an 
equation of the form 

tan X =p tan y 

which [PL Trig. Art 254] is 

X — y = J sin 2y + t J* sin 4y + &c. 
in which 

q =^ 

^ p + 1 

Applying this to the development of (82), we find, after divid- 
ing by sin V to reduce the terms of the series to seconds, 

9 — 9* = — -At;, Bin 2.9 — ;r^-^, sin ^9 — &c. (88) 

sm 1 ' 2 sm 1" 

in which 

;> — 1 1 — c» — 1 e 

? = 

p + 1 1— e»+l 2— 6» 

Employing Bessel's value of r, we find 

^ = 690".65 ^ — = — r.l6 

sin 1" 2 sin 1" 

and, the subsequent terms being insensible, 

^ — ^' = C00".65 sin 2f> — 1".16 sin 4f> (88*) 

by which ^ — ^' is readily computed for given values of f>. Its 
value will be found in our Table III. Vol. 11. for any given 
value of 9. 

Example. — Find the reduced latitude when ip = 85®. We find 
by (83), or Table HI., 

^ _ ^ = 648".25 = IC 48".25 

and hence the reduced or geocentric latitude 

9' = 34° 49' 11".75 



82. To find the radius of the terrestrial spheroid for a given latitude. 

p = the radius for the latitude ^ = AC. 
Vfe have 

To express x and y in terms of f , we have from the equation of 
the ellipse and its diflferential equation, after substituting 1 — c* 

for -19 

3? H ^ = a« 

i?. = (1 — c«) tan 9> 


from which by a simple elimination we find 

a cos cp 

and hence 

l/(l — €* sinV) 

(1 — c^ a sin^ 

l/(l — e" sinV) 

/ r 1 — 2 6* siii*f> + e* BJnV ] 

''""^ VL l~6«sinV J 


by which the value of p may be computed. The logarithm of 
/>, putting a = 1, is given in our Table m. Vol. 11. 

But the logarithm of p may be more conveniently found by a 
series. If in (84) we substitute 

e = \ —p 

sin's? = J (1 — cos 2f) 
we find, putting a = 1, 

^ /[ !+/* + (!-/*) cos 2y> 1 

^Ll+/' + (l-/')C08 2j 


+ /' + (l-/')C08 2v,. 


Now (PI. Trig. Art. 260) if we have an expression of the form 

X=|/(l + m' — 2mcos C) {A) 


we have, if -Jf = the modulus of the common system of loga- 

1 TT umI n i tn* COB 2C , m»cos3C , . \ ,^ 
log Jr=— -MImcos 0-\ 1 1- &c. I (5> 

by which we may develop the logarithms of the numerator and 
denominator of the above radical. 
Hence we find 

log p = log + Ml (m — m!) cos 2^ cos 4f 

-| cos 6^ — &c. j 

in which we have put for brevity 

1-/' , 1-/ 

m = — m == - 

1+/' 1+/ 

Restoring the value of /= |/(1 — 6*) and computing the 
numerical values of the coefficients, we find 

log p = 9.9992747 + 0.0007271 cos 2?) — 0.0000018 cos 4^ (85) 

as given by Encke in the Jahrbuch for 1852. 

The values of p and <p' may also be determined under another 
form which will hereafter be found useful. 

We have in Fig. 11, p sin (p^ =^y^ p cos ^' = x, or 

a (\ — e*) sin «p 
p Bm 0' = — ^^- : 

»/(l-e'8inV) . (gg) 

a cos I \ ^ 

p cos = 

^ ^ |/(1 — e» sin' if) 

which may be put under a simple form by introducing an auxi- 
liary 4'? so that 

sin 4 = e sin ^ 
p sin ^' = a (1 — e*) sin ^ soc 4 \ (87) 

^ cos (p' =: a cos ^ soc 4 

"We can also deduce from these, 

p sin (^ — ^*) = I fle* sin 2f sec 4 
/> cos (^ — ^) = a cos 4 

} (88) 

NORMAL. 101 

Hence, also, the following: 

, = aJ( r^ ^) (89) 

\ \ COS ^' COS (^ — /) / ^ "^ 

83. To find the length of the normal terminating in the axis^ for a 
given latitude. 


N= the normal = AO (Fig. 11), 

we have evidently 

N= ^.^^ = — — ? (90) 

COS y> ^(1 — e* sin' ^) 

or, employing the auxiliary 4' of the preceding article, 

N= a sec 4 

84. To find the distance from the centre to the intersection of the 
normal with the axis. 

Denoting this distance by ai (so that i denotes the distance 
when a = 1), we have in Fig. 11, 

ai = CO 
and, from the triangle ACOy 

P sin (cp — «>') 

at = — ^-^ 

cos <p 

or, by (88), 

ae* sin ^ , . ,_,^ 

ax = -77^ ., . , , = ae^ sm cp sec 4 (91) 

l/(\ — e* sin* tp) ^ ^ ^ ^ 

85. To find the radius of curvature of the terrestrial meridian for a 
given latitude. — Denoting this radius by if, we have, from the dif- 
ferential calculus, 

where we employ the notation D^y^ JD/ y to denote the first 
and second dilterential coefficients of y relatively to x. We 
have from the equation of the ellipse 

-TV h^ X ^ 6* 

^xy = --;T.-^ J^:y = 

a* y * ^ a^rf 



Observing that 6^ = a* (1 — e^), we find, by substituting the values 
of X and y in terms of ^ (p. 99), 

^ = 71^^ (92) 

(1 — e* sin' ^)t 

Example. — ^Find the radius of curvature for the latitude of 

Greenwich, <p = 51° 28' 38".2, taking a = 6377397 metres. We 


R = 6373850 metres. 

86. Abnormal deviations of the plumb line. — Granting the geo- 
metrical figure of the earth to be that of an ellipsoid of revolu- 
tion whose dimensions, taking the mean level of the sea, are as 
given in Art. 80, it must not be inferred that the direction of the 
plumb line at any point of the surface always coincides precisely 
with the nonnal of the ellipsoid. It would do so, indeed, if the 
earth were an exact ellipsoid composed of perfectly homoge- 
neous matter, or if, originally homogeneous and plastic, it has 
assumed its present form solely under the influence of the 
attraction of gravitation combined with the rotation on its axis. 
But experience has shown* that the phimb line mostly deviates 
from the normal to the regular ellipsoid, not only towards the 
north or south, but also towards the east or west ; so that the 
apparent zenith as indicated by the plumb line difters from the 
true zenith corresponding to the normal both in declination and 
right ascension. These deviations are due to local irregularities 
both in the figure and the density of the earth. Their amount is, 
however, very small, seldom reaching more than 3" of arc in 
any direction. 

In order to eliminate the influence of these deviations at a 
given place, observations are made at a number of places as 
nearly as possible symmetrically situated around it, and, as- 
suming the dimensions of the general ellipsoid to be as we have 
given them, the direction of the plumb line at the given place is 
deduced from its direction at each of the assumed places (by 

♦ U.S. Coast Surrey Report for 1863, p. 14» 


the aid of the geodetic measures of its distance and direction 
from each) ; or, which is the same thing, the latitude and longi- 
tude of the place are deduced from those of each of the assumed 
places : then the mean of all the resulting latitudes is the geodetic 
htiiude, and the mean of all the resulting longitudes is the geodetic 
longitude^ of the place. These quantities, then, correspond as 
nearly as possible to the true normal of the regular ellipsoid ; 
the geodetic latitude being the angle which this normal makes 
with the plane of the equator, and the geodetic longitude being 
the angle which the meridian plane containing this normal 
makes with the plane of the first meridian. The geodetic lati- 
tude is identical with the geographical latitude as we have defined 
it in Art. 81. 

The astronomical latitude of a place is the declination of the 
apparent zenith indicated by the actual plumb line ; but, unless 
when the contrary is stated, it will be hereafter understood to be 
identical with the geographical or geodetic latitude. 

It has recently been attempted to show that the earth differs 
sensibly from an ellipsoid of revolution;* but no deduction of 
this kind can be safely made until the anomalous deviations of 
the plumb line above noticed have been eliminated from the 



87. The places of stars given in the Ephemerides are those in 
which the stars would be seen by an observer at the centre of 
the earth, and are called ^eoeew^nc, or true, places. Those observed 
ftt)m the surface of the earth are called observed, or apparent, 

It must be remarked, however, that the geocentric places of 
the Ephemeris are also called apparent places when it is intended 

♦ See Attr. Naeh. No. 1308. 


to distinguish them from mean places, a distinction which will 
be considered hereafter (Chap. XI.). 

It will also be noticed that we frequently use the terms true 
and apparent as relative terms only ; as, for example, in treating 
of the eftect of parallax, the place of a star as seen from the 
centre of the earth may be called true, and that in which it 
would be seen from the surface of the earth were there no 
atmosphere, may in relation to the former be called apparent; 
but in considering the effect of refraction, the star's place as it 
would be seen from the surface of the earth were there no atmo- 
sphere may be called true, and the place as affected by the re- 
fraction may in relation to the former be called apparent; and 
similarly in other cases. 


88. The parallax of a star is, in general, the difference of the 
directions of the straight lines drawn to the star from two different 
points. The difference of direction of two straight lines being 
simply the angle contained between them, we may also define 
parallax as the angle at the star contained by the lines drawn to 
the two points from which it is supposed to be viewed. 

In astronomy we frequently use the term parallax to express 
the difference of altitude or of zenith distance of a star seen 
from the surface and the centre of the earth respectively; 
and, in order to express parallax in respect to other co-ordi- 
nates, proper qualifying terms are added, as " parallax in decli- 
nation," &c. 

Assuming (at first) the earth to be a sphere, let -4, Fig. 12, be 

the position of the observer on its surface, 
C* the centre, (LIZ the vertical line, and S vl 
star within a measurable distance OS' from 
the centre. AJl^ a tangent to the surface 
at A, and (V/, parallel to it, drawn through 
the centre, may each be regarded as lying 
in the plane of the celestial horizon (note, 
p. lU). The true or geocentric altitude of 
the star above the celestial horizon is then 
the angle iSY7/, and the apparent altitude is 
the angle SAJP. In this case the directions of the star from C 
and from -1 are compared with each other by referring them to two 


lines which have a common direction, i.e. parallel lines. But a 
still more direct method of comparison is obtained by referring 
them to one and the same straight line, as CAZ, Z being the 
zenith. We then call ZCS the true and ZAS the apparent 
zenith distance, and these are evidently the complements of the 
tnie and apparent altitudes respectively. 
In the figure we have at once 


that is, the parallax in zenith distance or altitude is the angle 
at the star subtended by the radius of the earth. When the star 
is in the horizon, as at H\ the radius, being at right angles to 
AH\ subtends the greatest possible angle at the star for the same 
distance, and this maximum angle is called the horizontal parallax. 
The equatorial horizontal parallax of a star is the maximum angle 
subtended at the star by the equatorial radius of the earth. 

89. To find the equatorial horizontal parallax of a star at a given 
distance from the centre of the earth. 


r = the equatorial horizontal parallax, 

J = the given distance of the star from the earth's centre, 

a = the equatorial radius of the earth, 

we have from the triangle CAH' in Fig. 12, if CA is the 
equatorial radius. 

« (93) 

sin r = _ 

The value of n given in the Ephemeris is always that which is 
given by this formula when for J we employ the distance of the 
star at the instant for wliich the parallax is given. 

90. To find the parallax in altitude or zenith distance^ the earth being 
regarded as a sphere. 


Z =z the true zenith distance = ZCS (Fig. 12), 

C' = the apparent zenith distance = ZAS, 
p z= the parallax in alt. or z. d. = CSA, 


The triangle SAC gives, observing that the angle SAC 
=; 180° — C', 

sin /> a 

- — ^ = -7 = Sin r 
sin ; J 

sin j> = sin (C — Z) = sin r sin C' (94) 

If we put 

A == the trae altitude, 

A' = the apparent altitude^ 

then it follows also that 

sin /) = sin (A — A ') = sin r cos A' (95) 

Except in the ease of the moon, the parallax is so small that wo 
may consider ;r and p to be proportional to their sines [PL Trig. 
Art. 55] ; and then we have 

/) = r sin r =^ ^ cos A' (96) 

Since when f ' = 90° we have sin ^' = 1, and when f ' = 0, sin 
^' =z 0, it follows that the parallax is a maximum when the star 
is in the horizon, and zero when the star is in the zenith. 

Example. — Given the apparent zenith distance of Venus, 
f' = 6-4° 43', and the horizontal parallax ;r^20".0; find the 
geocentric zenith distance. 

log r 1.3010 
C = 64^ 43' 0"0 log 8in :' 9.9563_ 

p= 18.1 log/) 1.2573 

C = 04 42 41.9 

Wlicn the true zenith distance is given, to compute the paral- 
lax, we may first use this true zenith distance as the apparent, 
and tin<l an a|>proximate value of p by the formula p = z sin J; 
then, taking tlie ai>pr()ximate value of C' -- C ~ /^» ^^'^* compute a 
more exact value of p by the fonnula (94) or (OG). This second 
a|)pr<)ximati()n is unnecessary in all cases exce|>t that of the 
moon, and the parallax of the moon is so great that it becomes 
necessary to take into account the true fii^ure of the earth, as in 
the following more general investigation of the subject. 

91. Tn consefjuence of the spheroidal figure of the earth, the 
vertical line of the observer docs not pass through the centre, 
and therefore the geocentric zenith distance cannot be directly 


referred to this line. If, however, we refer it to the radius drawn 
from the place of observation (or CAZ'y Fig. 11), the zenith dis* 
tanee is that measured from the geocentric zenith of the place; 
whereas it is desirable to use the geographical zenith. Hence 
we shall here consider the geocentric zenith distance to be the 
angle which the straight line drawn from the centre of the earth 
to the star makes with the straight line drawn through the centre 
of the earth parallel to the vertical line of the observer. These two 
vertical lines are conceived to meet the celestial sphere in the 
same point, namely, the geographical zenith, which is the 
common vanishing point of all lines perpendicular to the plane 
of the horizon. Thus both the true and the apparent zenith 
distances will be measured upon the celestial sphere from the 
pole of the horizon. 

The azimuth of a star is, in general, the angle which a vertical 
plane passing through the star makes with the plane of the meri- 
dian. When such a vertical plane is drawn through the centre 
of the earth, it does not coincide with that drawn at the place of 
observation, since, by definition (Art. 3), the vertical plane passes 
through the vertical line, and the vertical lines are not coincident. 
Hence we shall have to consider a parallax in azimuth as well as 
in zenith distance. 

92. To find the parallax of a star in zenith distance and azimuth 
tchai the geocentric zenith distance and azimuth are given y and the earth 
is regarded as a spheroid.*^ 

Let the star be referred to three co-ordinate planes at right 
angles to each other : the first, the plane of the horizon of the 
observer; the second, the plane of the meridian; the third, the 
plane of the prime vertical. Let the axis of x be the meridian 
line, or intersection of the plane of the meridian and the plane 
of the horizon ; the axis of y, the east and west line ; the axis 
of ^, the vertical line. Let the positive axis of x be towards the 
south; the positive axis of y, towards the west; the positive 
axis of Zy towards the zenith. Let 

J' = the distance of the star from the origin, which is 

the place of observation, 
C' = the apparent zenith distance of the star, 
A' = the apparent azimuth " " " 

* The inyestigation which follows is nearly the some as that of Olbebs, to whom 
the method itself is due. 


then, 0^ y\ z' denoting the co-ordinates of the star in this system, 
we have, by (39), 

of = A* sin C* cos A 
y' = J' sin C' sin A 
2' = J' cos C' 

Again, let the star be referred by rectangular co-ordinates to 
another system of planes parallel to the former, the origin now 
being the centre of the earth. In the celestial sphere these 
planes still represent the horizon, the meridian, and the prime 
vertical. K then in this system we put 

J = the distance of the star from the origin, 
C = the true zenith distance of the star, 
A = the true azimuth " " 

and denote the co-ordinates of the star m this system by x, y, 
and z, we have, as before, 

a: = J sin C cos A 
y = J sin C sin A 
z =z J cos C 

Now, the co-ordinates of the place of observation in this last 
system, being denoted by a, 6, c, we have 

a =^ /o sin (^ — ^') 6 = c = p cos (^ — f ') 

in which /> = the earth's radius for the latitude f of the place of 
observation, and (p' is the geocentric latitude, f — <f' being the 
reduction of the latitude. Art. 81 ; and the formula? of transforma- 
tion from this second system to the first arc (Art. 33) 

X = xf -{- a y =: y' -^ b z = z' -{■ c 

or, x^ z= X — a y' = y — ^ z' = z — c 

whence, substituting the above values of the co-ordinates, 

J' sin r cos A^ = J sin C cos A — /> sin (^ — f ') ^ 

J' sin :' sin A' =^ J sin : sin A > (07) 

J' cos C* = J cos C — p cos (^ — s?') J 

which arc the general relations between the true and apparent 
zenith distances and azimuths. All the quantities in the second 
members being given, the first two equations determine J'sin ^', 
and A '; and then from this value of J'sin ^', and that of J'cos ^' 
given by the third equation, J' and J' are determined. 

PARALLAX. . 109 

But it is convenient to introduce the horizontal parallax 
instead of J. For, if we put the equatorial radius of the earth 

= 1, we have 


sm TT = — 

and hence, if we divide the equations (97) by J, and put 


we have 

/ sin ^ cos A' = sin C cos A — /o sin tt sin (jp — <p') \ 

f sin C' sin A' = sin C sin A v (98) 

/ cos C' = cos C — p Am: cos (^ — /) j 

To obtain expressions for the difference between ^ and f ' and 
between A and A'^ that is, for the parallax in zenith distance 
and azimuth, multiply the first equation of (98) by sin -4, the 
second by cos A^ and subtract the first product from the second ; 
again, multiply the first by cos -4, the second by sin -4, and add 
the products: we find 

/ sin C sin {A' — A) = /> sin r sin (f> — <p') sin A 1 

/ sin C' cos (A' — A) = sin C — p sin r sin {<p — <p') cos A / ^ ^ 

Multiplying the first of these by sin J (A' — A), the second by 
cos \ (A' —A), and adding the products, we find, after dividing 
the sum by cos \ (A' — A), 

cos } (A' + A) 

/ sin C' = sin C — p sin tt sin (^ — ^') 

cos 1 (A' — A) 

which with the third equation of (98) will determine f '. 

If we assume x such that 

^ / fN cos J (A' + A) ^^ 

tan p' = tan (sP — /) ry^T -r (100) 

cos } (A' — A) 

we have the following equations for determining f ' : 

/ sin C' = sin C — p sin - cos (^ — ^') tan y 
f cos C' = cos C — /o sin TT cos (^ — ^') 

which, by the process employed in deducing (99), give 

X - rt^ ^ • / /N sin (C — r) 

/ sm (r — C) = ^ s^^ ^ cos (jp — /) - ^ ^ 

} (101) 

/ cos (C' — C) = 1 — /> sin ir cos (sp — 9') 

cos p' 

cos (C — f) 
cos p' 


110 . PARALLAX. 

By multiplying the first of these by sin J (^' -— ^), the second 
by cos I (^' — f ), and adding the products, we find, after dividing 
by cos 4 (C' - C), 

/ o sin r cos {if ~ y') cos [} (C + ~ r] 
cos p' cos 1 (C' — C) 

or multiplying by J, 

J' = J — P ^^^ ^^ — ^') ^^^ 1^^ (^' + ^)-~ ^3 (108) 

cos Y cos J (C* — C) 

The equations (99) determine rigorously the parallax in 
azimuth ; then (100) and (102) determine the parallax in zenith 
distance, and (103) the distance of the star from the observer. 

The relation between A and J' may be expressed under a more 
simple form. By multiplying the first of the equations (101) by 
cos x^ the second by sin x^ the difterence of the products gives 

A' = J ^ILSL^IZI) (104) 

sin (C -^r) 

93. The preceding formulae may be developed in series. 

p&inn si n (y> — fQ 
m = ; — 

sm C 

then (99) become 

/ sin C sin (A' — ^) = m sin C sin A 

f sin C cos (A' — A) = sin C (1 — 7n cos A) 


^ A, jv ^ sin A ,,y*-x 

tan (A' — A) = (105) 

1 — m cos A 
and therefore [PL Trig. Art. 258], A' — A being in seconds, 

., . msinA , m*8in2A m'sinSA , . ,^^. 

A' — A = H H + &c. (106) 

sin 1" ^ 2 sin 1" ^ 3 sin 1" ^ ^ ^ 

To develop j^ in series, we take 

, , ,^ cos [A + iU'- ^)] 
tan X = tan (^p — f ') ?^ — -— ^ — ^ 

cos } (il — A) 
= tan (v> — ^') [cos A — sin A tan } (A' — A)] 

whence, by interchanging arcs and tangents according to the 


formulse tan~^ ^ = y — J ^ + &c., tan x = x + ^s^ + kc. [PI. 
Trig. Arts. 209, 218], 

/ #N A (sf> — ^')V sin r sin* ^ sin 1" , ^ .,^_ 

2 sin C 

where the second term of the series is multiplied by sin 1" 
because y and <p — ^' are supposed to be expressed in seconds. 
Again, if we put 

/t) sin rr cos (^ — ^) 

cos / 
we find from (102) 

taD(r-C)= """(g-r) (108) 

^ "^ 1 — nco8(C — r) 

whence, f ' — f being in seconds, 

nsinC^r) . n'sin2(C~r) . n>8in3(C-~r) . j^^^ ,109) 
sinl" ^ 2 8inl" ^ 3 sin 1" ^ ^ ^ 

Adding the squares of the equations (102), we have 
/* = ( jV = 1 — 2 n cos (C — r) + n« 

whence [equations {A) and (jB), Art. 82] 

log J' = log J — Jf (n cos (C — r) + ^* ^^^ ^ (^ ~" ^) . ^. &c.) (110) 

where -Jf =the modulus of common logarithms. 

94. The second term of the series (107) is of wholly inappre- 
ciable eflect ; so that we may consider as exact the formula 

;^ = (^ — /) cos A (111) 

and the rigorous formulae (105) and (108) may be readily com- 
puted under the following form : 


. - - p sin r sin (c> — c>') cos A 
sm^ = m cos -4 = '- ~ ^^ 

sm C 
then ^ (112) 

tan(-4'— -4) = ^5_^-^?iLd == tan * tan (45° + J *) tan il 



Bin d' = n cos (C — r) = — — ^^^ -^ 


. ,w ^. sin 1^' tan (C — r) ) (^^^) 

tan (r — = ; . '^ 

1 — Bin * 
= tan t>' tan(45*> + } 6')tan (C — r)y 

Example. — In latitude ^ = 38° 59', given for the moon, A = 
320^ 18', ^ -: 29^ 30', and ;r = 58' 37".2, to find the parallax in 
azimuth and zenith distance. 

We have (Tahle III.) for f = 38° 59', ip — ip' = 11' 15", log p 
= 9.999428: hence by (111) ;' = 8'39".3 and f — r = 29° 21' 
20".7 ; vdth which we proceed by (112) and (113) as follows : 

logpsinTT 8.28118 log p sin tt 8.281179 

log sin (^ — f ) 7.51488 log cos (^ — f ) 9.999996 

log coseo C 0.80766 log sec y 0.000001 

log cos A 9.88615 log cos (C — >) 9.940818 

^ = 18", log sin ^ 5.98987 ^'= 51' 1".5, log sin i^ 8.171491 

logtani? 5.93987 log tan t^ 8.171589 

log tan (450 -f J ,9) 0.00004 log tan (45o 4- } ^) a006446 

log tan A n9.91919^ log tan (C — y) 9.760087 

log tan {A' — A) n5.8o910 log tan (C — C) 7.928072 

A' — A ---. — 14".91 C — C = 29' 7^.79 

A = 820° 17' 4ry'.09 ^=290 59' 7^.79 

It 18 evident that we may, without a sacrilice of accuracy, 
omit the factors cos {f — (p') and cos y in the computation of sin &\ 

If we neglect the compression of the earth in this example, 
we find by (94) C — (: = 29' 17".9, which is 10" in error. 

95. To find the parallax of a star in zenith distance and azimuth 
when the apparent zniith distance and azimuth are given^ the earth 
being regardM as a spheroid. 

If we multiply the fii*st of tlie equations (101) by cos f' and the 
second by sin ^', the difterence of the products gives 

a;« rr» r\ _ P ^'" ^ ^Q^ (^ ~ y') ^^'" (^ ~ r^ 
sin (^, — ^y — 


for wliidi, since cos [if — ^') and cos y are each nearly equal to 
unity, we may take, without sensible error, 

sin (::' — :)=/> sin TT Bin (C — r) (H*) 


in which y has the value found by (111), or, with sujfieient accu- 
racy, by the formula 

r = (sp — ?»') cos ^' (115) 

Again, if we multiply the first of the equations (98) by sin A' 
and the second by cos A\ the difference of the products gives 

sin (^' -A-) = PBi""«in(y-/)8in^' 

sin C 

to compute which, ^ must first be found by subtracting the value 
of the parallax ^' — ^, found by (114), from the given value of ^'. 

Example. — ^In latitude tp = 38° 59', given for the moon A^ = 
320° 17' 45".09, C' = 29° 59' 7".79, tt = 58' 37".2, to find the 
parallax in zenith distance and azimuth. 

We have, as in the example Art. 94, ^ — ^' = 11' 15", log p 
= 9.999428, r = {v — v') cos A' = 8' 39".3, C' — r = 29° 50' 28".5 ; 
and hence, by (114) and (116), 

log p sin r 8.281179 log p sin r 8.28118 

log sin (C — r) 9.696879 log sin (^ — ^') 7.51488 

log sin (C — C) 7.928058 log sin A' w9.80588 

C — C = 29' 7".79 log cosec C 0.80766 

C = 29° 80' 0" log sin {A' — A) n5.85910 

A' — A = — 14".91 
A = 820° 18' 0" 

agreeing with the given values of Art. 94. 

96. For (lie planets or the sun^ the following formulae are always 
BuflBiciently precise : 

c'-c=^;r6in(:'-r) 1 

A — A= pit sin (^ — 9?') sin A' cosec C' j v. ^« ; 

and in most cases we may take C ~Z^^'^ ®^^^ C') ^^^ -^' — ^ = 0. 
The quantity pr: is frequently called the reduced parallax^ and 
t: — ^;r = (1 — p):: the reduction of the equatorial parallax for the 
given latitude ; and a table for this reduction is given in some 
collections. This reduction is, indeed, sensibly the same as the 
correction given in our Table XIII., which will be explained 
more particularly hereafter. Calling the tabular correction a;:, 
we shall have, with sufficient accuracy for most purposes, 

pn =^ n — A;r 
VouL— 8 



97. The preceding methods of computing the parallax enable 
us to pass directly from the geocentric to the apparent azimuth 
and zenith distance. There is, however, an indirect method 
which is sometimes more convenient. This consists in reducing 
both the geocentric and the apparent co-ordinates to the point m 
xchich the vertical line of the observer intersects the axis of the earth. I 
shall briefly designate this point as the point (Fig. 11). 

We may suppose the point to be assumed as the centre of 
the celestial sphere and at the same time as the centre of an 
imaginary terrestrial sphere described with a radius equal to the 
normal OA (Fig. 11). Since the point is in the vertical line of 
the observer, the azimuth at this point is the same as the appa- 
rent azimuth. If, therefore, the geocentric co-ordinates are first 
reduced to the point 0, we shall then avoid the parallax in 
azimuth, and the parallax in zenith distance will be found by the 
simple formula for the earth regarded as a sphere, taking the 
normal as radius. 

Since the point is in the axis of the celestial sphere, the 
straight line drawn from it to the star lies in the plane of the 
declination circle of the star; the place of the star, therefore, as 
seen from the point 0, difters from its geocentric place only in 
declination, and not in right ascension. We have then only to 
find the reduction of the declination and of the zenith distance 
to the point 0. 

Ist. To reduce the declination to the point O. — ^Let 
PP', Fig. 13, be the earth's axis; C'the centre; 
the point in which the vertical line or normal 
of an observer in the given latitude f meets the 
axis; S the ytar. We have found for CO the 
expression (Ail. 84) 

CO = ai 

in which a is the equatorial radius of the earth, 


c* sin ^ 

Fig. 13. 

I = 

|/(1 — c* sin* f) 


J =:^ the star's geocentric distance 

Jj ~z the star's distance from the point 

o -.- the geocentric declination 

^^ — the declination reduced to the point 




— PCS, 




then, drawing SB perpendicular to the axis, the right triangles 
SCB and SOB give 

J, sin a, == J sin ^ + ai \ 

J, cos a, = J COB ^ / ^^ •' 

which determine J^ and 3^. Prom these we deduce 

Jj sin (^j — ^) = ^i cos 9 

Jj cos (^4 — ^) = J -f- flt sin ^ 

} (119) 

which determine J^ and the reduction of the declination. If we 
divide these by J, and put 

^ 4i .a 

f^ = i Bm^ = - 

in which n denotes, as before, the equatorial horizontal parallax, 
they become 

/j sin (^4 — S) =:% sin ^ cos d 
/j cos (d^ — ^) = 1 -(- t sin TT sin ^ 

^ , . .. i sin TT cos d 

tan (a, — ^) = ^ , . . 7—^ 

1 -f t sm r sm d 

or in series [PI. Trig. Art. 257], 

, ^ t sin TT cos ^ ft sin «)' sin 2 ^ . ., 
o. — tf = -^ V- &c. 

sinl" 2sinr' ^ 

But since the second term of the series involves f* and conse- 
quently e*, and this is further multiplied by the small factor sin^ ;r, 
the term is wholly inappreciable even for the moon; and, as 
the first term cannot exceed 25" in any case, we shall obtain ex- 
treme accuracy by the simple formula 

^1 — ^ = t ff cos a (120) 

The value of A^ is found from (119), by the same process as 
was used in finding J' in (103), to be 

. . f 1 , . . sin J (a, + ^) 1 
* \ ^ cosJ(a, — a)/ 

or, on account of the small difterence between d^ and ^, 

J, = J (1 + i sin TT sin ^ (121) 



As ^1 — ^ is 80 small, it may be accurately computed with 
logarithms of four decimal places, and it will be conveuient to 
substitute for i the form 

in which 

i = A sin ^ 

A = 

|/(1 — e* sin* f ) 

The value of log A may then be taken from the following 
table with the argument ^ = the geographical latitude 























•We shall then compute 8^ — 3 and Jj under the following 
forms : 

9^ — d = A t: sin ^ cos d 

J^ = J (1 -\- A sin K sin f> sin d) 

} (122) 

K the value of z^ has been found as below, we may take 

d^ — d = e* r^ sin f cos d 





To find the p^irallax in zenith distance for the point 0. — Let 
ZAOy Fig. 14, be the vertical line of the ob8er\'er at 
A. The normal ylO terminating in the axis being 
denoted by iV, we have, by (90), 


N = 


|/(I — e* sin* ^) 

But if in (84) we write c* sin* <f for c* sin' f , we have 

p = a |/(1 — e* sin' ^) 

and this value is sufficiently accurate for the compu- 
tation of the parallax in all cases. If then we put 
a — ly we have 



K now in the vertical plane passing through the line ZO and 
the star S we draw SB perpendicular to OZ, and put 

Cj = the zenith distance at O = SOZ 
C' = the apparent zenith dist. = SAZ 

the triangles 0/SB, ASB give 

J' COB r = J. COB C- I I ^j23) 

J' sin C' = ^1 sin d J 

Dividing these equations by Jp and putting 

J' 1 

— =/ sin TT, == 

A ' Hi 

they become 

/i cos C' = cos Cj — sin ^^ 
/j sin C' = sin Ci 

from which we deduce 

/j sin (C' — Ci) = sin r^ sin C^ 

/j cos (C' — Cj) = 1 — sin TTj cos Cj 

tan (C - C.) = , «'""»»'"^« (124) 

1 — sm TTj cos Ci 

and in series, 

sin ;r, sin C, sin' tt, sin 2 C, ^ 
^ ^*- sinl" + 2sinl" ^ ^' ^^^^^ 

Or, rigorously, 

sin t* = sin n^ cos C, 
tan (C — Ct) = tan ^ tan (45^ + J ^) tan C^ 

To find ^Ti we have 


sm 7c, = — = 

or sm TT, = 

* jo Jj /» J (1 -f- ji sin TT sin ^ sin ^) 

sin 9r 

} (126) 

p(l + A sin TT sin ^ sin S) 



But this very precise expression of tt^ will seldom be required: 
it will generally suffice to take 

sm ffj = 


or TTj 



which will be found to give the correct value of tt^^ even for the 
moon, within 0".2 in every case. Wliere this degree of accu- 
racy suffices, we may employ a table containing the correction 
for reducing n to t:^^ computed by the formula 



Table XTEL, Vol. 11., gives this correction with the arguments 7t 
and the geographical latitude ip. Taking the correction from 
this table, therefore, we have 

JTj = TT 4- Ajt (128) 

Zd, To compute the parallax in zenith distance for the point O when 
the apparent zenith distance is (jiven. 

Multiplying the first equation of (123) by sin J^', the second by 
cos ^', and subtracting, we find 

8in(r — C,) = — sinC 
or sin (C — Cj = sin r^ sin C (129) 

If we denote the apparent altitude by A' and the altitude 
reduced to the point hy Aj, this equation becomes 

sin (Aj — A') = sin r^ cos W (130) 

Example. — In Latitude ip = 38° 59', given the moon's hour 
angle t =- 341° 1' 3«".85, geocentric declination d = + W 39' 
24".r>4, and the equatorial horizontal parallax ;: — 58' 37".2, to 
find the apparent zenitli distance and azimuth. 

The geocentric zenith distance and azininth, computed from 
tliese data by Art. 14, are Z '- 29° 30', A ■--- 320° 18', whicli are 
the values employed in our example in Art. 94. To compute 


by the method of the present article, we first reduce the declina- 
tion to the point by (122), as follows : 

For ^ = 38^ 59' log A 7.8250 

IT = 3517".2 log n 3.5462 

log sin f 9.7987 

^ = 14*> 39' 24".54 log cos a 9.9856 

d^—d= 14 .31 \og(d^—d) 1.1555 

a, = 14** 39' 38".85 

With this value of 3^ and t = 341° V 36".85, the computation 
of the zenith distance and azimuth by Art. 14 gives for the 
point O 

C, = 29^ 29' 47".67 A, = 320^ 17' 45".09 

and this value of J.^ is precisely the same as A' found in Art. 94, 
as it should be, since the azimuth at the point and at the 
observer are identical. 

We find from Table XIH. a;: = 4''.6, and hence ;ri= 58' 37".2 
+ 4''.6 = 58' 41".8; and then, by (126), 

log sin TT^ 8.23232 
log cos C, 9.93971 

d = 51' 5" log sin d 8.17203 

log tan »9 8.17208 

log tan (45° + i »>) 0.00645 

C, = 29*> 29' 47".67 log tan C^ 9.75258 

r — C, = 29 20 .03 log tan (C— CJ 7.93111 

C' = 29° 59' 7".70 

agreeing with the value found in Art. 94 within 0".09. If we 
had computed t:^ by (127), the agreement would have been exact. 

98. To find the parallax of a star in right ascension and declination 
when its geocentric right ascension and declination are given. 

The investigation of this problem is similar to that of Art. 92. 
Let the star be referred by rectangular co-ordinates to three 
planes passing through the centre of the earth : the first, the 
plane of the equator ; the second, that of the equinoctial colure ; 
the third, that of the solstitial colure. Let the axis of x be the 
straight line drawn through the equinoctial points, positive 
towards the vernal equinox ; the axis of y, the intersection of 


tlie plane of the solstitial eolure and that of the equator, positive 
towards that point of the equator whose right ascension is 90® ; 
the axis of z, the axis of the heavens, positive towards the nortL 


a = the star's geocentric right ascension, 
S = " " declination, 

J = " " distance, 

then the co-ordinates of tlie star are 

X = J cos d cos a 

y = J cos d Bin a 
2 = J sin ^ 

Again, let the star he referred to another system of planes 
parallel to the first, the origin being the observ^er. The vanish- 
ing circles of these planes in the celestial sphere are still the 
equator, the equinoctial eolure, and the solstitial eolure. Let 

a' = the star's observed right ascension, 

if = " " declination, 

J' = " distance from the observer, 

where by observed right ascension and declination we now mean 
the values which diiier from the geocentric values by the paral- 
lax depending on the position <>f the observer on the surface of 
the earth. The co-onlinates of the star in this svstem will be 

X' = J' cos tY cos a 

i/ =:z J' COS o sin a' 
y ^ J' sin <r 
Xow, if 

0= the sidereal timer- the right ascension of the observer's 

meridian at the instant of obscn'ation, 
^' = the reduced latitude of the place of observation, 
p =1 the radius of the earth for this latitude, 

then 0, if\ and o are the polar eo-onlinates of the observer, 
entirely analogous to a, o, and J of the star, so that the rectan- 
gular co-ordinates of the observer, taken in the first system, are 

a = /> cos ^''eos© 
b z= p ciw 9"' sin 
c -- p !jin y' 


and for transformation from one system to the other we have 

x' = x — a, !/ = y — ^f n^ = 2 — c. 

J' cos if cos o' = J cos d cos a — p COS f ' COS 

J' COS ^ sin o' = J cos 5 sin a — p cos ^ sin O y (131) 
J' sin ^ = J sin ^ — p sin f ' 

or, dividing by J, and putting as before 

J' 1 

/=_ sin»=-j- 

/ cos ^ cos a' = COS 9 COS a — p Bin 7C COS f ' COS 

/ COS ^ sin o' = COS ^ sin a — p sin ;r cos v>' sin y (132) 

/ sin ^ = sin ^ — ^o sin rr sin ^' 

From the first two of these equations we deduce 

/ cos ^ sin (a' — o) = /» sin TT cos f ' sin (o — 0) \ i qq 

/ cos ^ cos (a' — a) = COS d — p Bin t: COS f' COS (a — ^) ) ^ 

Mnltiplj-ing the first of these by sin J (a' — a), the second by 
cos i (a' — a), and adding the products, we find, after dividing by 
cos i {a' — a), 


- V - /» sm IT cos / cos rj (tt' + o) — 01 
/ cos ff = cos ^ — — , ,\ ^ — ■ — ^ =i 

cos 1 (a' — o) 

. tan ^' cos 1 (a' — a) ,- „ .^ 

**" ^ = COB a (,- + ,) - 0] (13^) 

then we have, for determining 5', 

/sin ^ = sin ^ — p sin rr sin ^ 

f cos ^ = cos J — /» sin rr sin f>' cot y 


sin (^ — r) 

} (135) 

/ sin (^ — ^) = P sin tt sin ^ ' 

sm ^ 

/ cos (^ — ^) = 1 — p sm TT sm 0' ^ ^ 

sm ^ 


J sm (^ — Y) ^ 

The equations (133) determine, rigorously, the parallax in right 


ascension, or a' — a ; (136) the parallax in declination, or 5' — J; 
and (137) determines J'. 

99. To obtain the developments in series, put 

P sin r cos tp' 

m = 

cos d 

then from (133) we have 

J. J' f V wi sin (a — 0) ,ioo\ 

tan (a' — a) = = 5^-p ^. (138) 

^ ^ 1 — m cos (a — 6) ^ 


m sin (a — 6) , m' sin 2 (a — O) , . ,^^^^ 


/> sin r sin <?' 

n = ; n 

sm Y 
we have from (136) 

/ V iN ^ sin (^ — y) 
tan (^ — d) = := ,, ^ , n40) 

^ '^ 1 — n cos {^d — ;') ^.^'v-/ 


sm 1 2 sui 1 ' ^ ^ 

100. The quantity a — is the hour angle of the star oast of 
the meridian. Acconling to the usual practice, we sliall reckon 
the hour angle towards the west, and denote it by /, or put 

t=e —a 

and then we shall write (138) and (140) as follows : 

. . ,. m sin t 

tan (a — o') = -— - 

1 — jn cos t 

tan (a - J-) ="*''" "1-*^- 

1 — n eos (J' — *i) 

Tlie rigorous computation will be conveniently performed by 
the followinsr forniulie: 

sin t> = m cos t = 


/9 sin ir cos ^' COS t 

COS d 

tan (a — a') = tan ^ tan (45® + } d) tan t 

tan «p' cos } (a — o') 

cos p + 1 (o — o')] t \ J 

. «/ X .N P sin iz sin «p' cos (y — S) 
sm ^' = n cos (j —- ^) = '- ^ ^ ^ 

sin Y 
tan (a — a') = tan ^ tan (45® + J ^) tan (y — 5) 

101. Except for the moon, the first terms of the series (139) 
and (141) will suffice, and we may use the following approxi- 
mations : 

pn COS <p' sin t 

a — a = 5 

COS d 

tan o' . ^^ ^^^ 

tanr = -;^ \ (143) 

^ __ ^ ^ joff sin f' sin (r — ^) 

sin ;^ 

If the star is on the meridian, we have / = 0, and hence 
X = if'y and 

d — ^ = /wr sin (f ' — d) 

Since in the meridian we have Z = <p — 3, it is easily seen 
that C' — C fo^^^d by (108) and d' — d found by (140) will then 
be numerically equal, or the parallax in zenith distance is numeri- 
caUy equal to the parallax in declination loJien the star is on the meri- 

102. To find the parallax of a star in right ascension and declination, 
when its obsei^ved right ascension and declination are given. 

Multiplying the first equation of (132) by sin a', the second 
by cos a', and subtracting one product from the other, we find 

. , -. p sin TT cos cp' sin (0 — o') 
sm (a — a ) = ^^ ^ 

cos d 

In like manner, from (135) we deduce 


• /. v\ /> Bin TT sin / sin (;- — ^) 
sm {d — dr) = ^ 

sin Y 

We have here — a' equal to the apparent or observed hour 
angle ; and hence, putting 

t'=e — a 


the computation may be made under the following form : 

. , ,, p sin TT cos cp' sin t' 

sm (a — o') = ^ 

^ ^ cos a 

tan / cos J (tt — o') 
tan r = ^^ ^ 

cos [f — J (a — a')] 

. / • vv /> sin r sin ^' sin (r — ^) 

sm (jS — o ) = ' 

sin Y 


In the first computation of a — a' we employ 5' for 8. The 
value of a — a' thus found is sufficiently exact for the compu- 
tation of -jf and o — 5'. With the computed value of 3 — d' we 
then find 3 and correct the computation of a — a'. 

Example. — Suppose that on a certain day at the Greenwich 
Observatory the right ascension and declination of the moon 
were observed to be 

tt' = 7* 41~ 20'.436 
^f = 15^ 50' 2r.66 

when the sidereal time was 

and the moon's equatorial horizontal parallax was 

n == 66' 57".5 
Required the geocentric right ascension and declination. 

We have for Greenwich ^ = 51° 28' 3ft".2, and hence (Table III.) 
^ - if' .-=- IV 13".r», ^' r. 51° 17' 24".r), log p = 9.9991134. The com- 
putation by (144) is then as follows: 



a' (in arc) 



1160 2(y 6''.54 
169 15 .80 


}(a-a') = 
</.-J(a — tt') = 

log 8€C [<* — J (a — 
log cos J (a — a') 
log tan ^' 

log tan y 

y = 

log p sin IT 
log sin ^' 
log Bin (y — d*) 
log cosec 7 

log sin (cJ — 6') 

6 — 6' = 
6 = 

53 54 58 .76 

14 55 .8 
58 89 58 

o')] 0.227819 


= 640 85' 58" 
r 48 45 80 



= -f 86' 55".24 
= 16O27'22".90 

log p sin IT 

log COS ^' 

log sin f 


log cos (T 




App. log sin (a — a') 7.938828 

Approx. a — a' = 29* 51".6 

(1) . . 7.922008 

log cos 6 9.981835 

log sin (a — a') 7.940173 

a — o'= + 29'57".23 

a =115050' 3".77 
= 7A43«20'.251 

103. For all bodies except the moon, the second computation 
will never affect the result in a sensible degree, and we may use 
the following approximations : 

, pn cos ^ sin f 

a — a 


cos ^ 
tan ^ 

cos f 

(m sin fff sin {y 




For the sun, planets, and comets, it is frequently more conve- 
nient to use the geocentric distance of the body instead of the 
parallax, or, at least, to deduce the parallax from the distance, 
the latter being given. This distance is always expressed in 
parts of the sun's mean distance as unity. K we put 

^r^ = the sun's mean equatorial horizontal parallax, 
J^ = the sun's mean distance from the earth, 

we have, whatever unit is employed in expressing J, J,,, and a, 

sm TT 


sm ff„ = — 




sm Tf = ~ Bin ita 

and when we take J^ = 1, 

am -K = 2. or r = -5. (146) 

According to Encke's detennination 

T^= 8".57116 log Tt^ = 0.93304 

Example. — ^Donati's comet was observed by Mr. James Fer- 
guson at Washington, 1858 Oct. 18, 6* 26* 2r.l mean time, 
and its observed right ascension and declination when corrected 
for refraction were 

o! = 236^ 48' 0".5 
^ = — T*> 36' 52".8 

The logarithm of the comet's distance from the earth was log A 
= 9.7444. Required the geocentric place. 

AVe have for Washington <p = 38° 53' 39". 3, whence, by Tabic 
in., log /o cos f ' = 9.8917, log/) sin f' = 9.7955. Converting the 
mean into sidereal time (Art. 50), we find = 19* 55* 16*.98. 
Hence, by (145) and (146), 


— 298° 49'.2 

log tan / 9.9038 


— 236 48.0 

log cos f 9.6713 


= 62 1.2 

log tan r 0.2325 



r — 69*» 39'.2 

log J 


r — ^= 67 16.1 



log />:r cos v>' 1.0803 log pTt sin ^ 0.9841 

log sin f 9.9460 log sin (r — ^) 9.9649 

log sec ^ 0.0038 log eosec r 0.0640 

log (a — a') 1.0801 log (a — ^) 1.0130 

a — a' = + 10".7 a ~ ^ = + 10".3 

Hence, for the geocentric place of the comet, 

a = 236° 48' 11".2 a = ~ 7° 36' 42".5 

104. Parallax in ladtufk ami longitude. — Formnhe similar to the 
above obtain for the parallax in latitude and longitude. We 


have only to substitute for and ^' (which are the right ascension 
and declination of the geocentric zenith) the corresponding 
longitude and latitude of the geocentric zenith (which will be 
found by Art. 23), and put X and ^ in the place of a and d. Thus, 
if I and b are the longitude and latitude of the geocentric zenith, 
the equations (1*43) give for all objects except the moon. 

, pT: cos b sin (/ — X) 

cos ^ 

tan b 

tanr = -^ ) (147) 

cos (/ — A) f \ J 

B^ff= ^^ ^'" ^ ^'" ^^ "~ ^^ 

sin Y 

In the same manner, the equations (131) may be made to 
express the general relations between the geocentric and the 
apparent longitude and latitude, and for the moon we can 
employ (142), observing to substitute respectively 

for o, 






quantities X, 






In all the formulfe, when we choose to neglect the compression 
of the earthy we have only to put ^ = ^' and p = l. 


105. General laws of refraction. — The path of a ray of light is a 
straight line so long as the ray is passing through a medium of 
uniform density, or through a vacuum. But when a ray passes 
obliquely from one medium into another of difterent density, it 
is bent or refracted. The ray before it enters the second medium 
is called the incident ray ; after it enters the second medium it is 
called the refracted ray; and the diiference between the directions 
of the incident and refracted rays is called the refraction. 

If a normal is drawn to the surface of the refracting medium 
at the point where the incident ray meets it, the angle which the 
incident ray makes with this normal is called the angle of inci- 
dence^ the angle which the refracted ray makes with the normal 
is the angle of refraction^ and the refraction is the difference of 
these two angles. 


Thus, if SAy Fig. 15, is an incident ray upon the surface BB' 

of a refracting medium, AC the refracted 
ray, 3IN the normal to the surface at A^ 
SAM is the angle of incidence, CAN is the 
angle of refraction ; and if CA be produced 
backwards in the direction AS\ SAS' is the 
refraction. An ol)ser\'er whose eve is at 
any point of the line AC will receive the 
rav as if it had come direct Iv to his eve 
without refraction in the direction S'ACj 
which is therefore called the apparmi 
direction of the rav. 

Xow, it is shown in Optics that this refraction takes place 
according to the following general laws: 

1st. AMien a ray of light falls upon a surface (of any form) 
which separates two media of different densities, the plane which 
contains the incident rav and the normal drawn to the surface 
at the jK:>int of incidence contains the refracted my also. 

2d. When the ray i>asses from a nirer to a denser medium, it 
is in general refracted toicards the normal, so that the angle of 
refraction is less than the angle of incidence ; and when the ray 
passes fn>m a denser to a rarer medium, it is refracted /row the 
normal, so that the angle of refraction is greater than the angle 
of incidence. 

3il. Whatever mav be the anirle of incidence, the sine of this 
angle bears a constant ratio to the sine of the corri'sjKinding 
anirle of retraction, st> lontr as the densities of the two media are 
constant. If a ray passes ont i>f a vacuum into a given meflium, 
the number exj»ressing this constant ratio is lalled the imhx of 
r»fr»t**t:..n for that medium. This index is always an improper 
fra«-t:*»ru being c«jual to the sine of the angle oi incidence divided 
bv thf -iric of the auirle oi refrartion. 

4::i. Wh»ii the ray passes fi\>m one medium into another, the 
sii;e- I'f th«- an*rlcs of incidence ami refraction art* n:*iiprocaUy 
ppi{«irt:oiial to the indices of refraction of the two meilia. 

1<». A-^r"hom''t**if rt'fracdon, — The ravs of liirht tmm a star in 
Ct»miiii: t«» the obser\er must pass thn>ugh the atmosphere which 
sum»u!nl> the earth. If the sji;ue between the ^ta^ and the 
upper limit of the atmosphere be reganled as a vaeuum, or as 
filled with a medium which exerts no sensible effect upon the 



direction of a ray of light, the path of the ray will be at first a 
straight line; but upon entering the atmosphere its direction 
will be changed. According to the second law above stated, the 
new medium being the denser, the ray will be bent towards th<^ 
normal, which in this case is a line drawn from the centre of the 
earth to the surface of the atmosphere at the point of incidence. 

The atmosphere, however, is not of uniform density, but is 
most dense near the surface of the earth, and gradually decreases 
in density to its upper limit, where it is supposed to be of such 
extreme tenuity that its first eftect upon a ray of light may be 
considered as infinitesimal. The ray is therefore continually pass- 
ing from a rarer into a denser medium, and hence its direction 
is continually changed, so that its path becomes a curve which 
is concave towards the earth. 

The last direction of the ray, or that which it has when it 
reaches the eye, is that of a tangent to its curved path at this 
point; and the difl:erence of the direction of the ray before en- 
tering the atmosphere and this last direction is called the astro- 
nomical refraction^ or simply the refraction. 

Thus, Fig. 16, the ray Se from a star, entering the atmosphere 
at e, is bent into the curve ccA 
which reaches the observer at A in 
the direction of the tangent S'A 
drawn to the curve at A. If CAZ 
is the vertical line of the observer, 
or normal at A^ by the first law of 
the preceding article, the vertical 
plane of the observer which con- 
tains the tangent AS' must also 
contain the whole curve Ae and 
the incident ray Se. Hence refrac- 
tion increases the apparent altitude 
of a star, but does not aftect its azi- 

The angle S'AZ is the apparent ze- 
nith distance of the star. The true zenith distance* is strictly the 
angle which a straight line drawn from the star to the point A 

Fig. 16. 

* By true xenith distance we here (and so long as we are considering only the 
effect of refraction) mean that which differs from the apparent lenith distance only 
by the refraction. 
Vol. L— » 


makes with the vertical line. Such a line would not coincide 
with the ray Se; but in consequence of the small amount of the 
refraction, if the line Se be produced it will meet the vertical 
line AZ at a point so little elevated above A that the angle 
which this produced line will make with the vertical will difier 
very little from the true zenith distance. Thus, if the produced 
line Se, be supposed to meet the vertical in 6', the difterence 
between the zenith distances measured at 6' and at A is the 
parallax of the star for the height Ab'y and this difference can l)c 
appreciable only in the case of the moon. It is therefore usual 
to assume Se as identical with the ray that would come to the 
observer directly from the star if there were no atmosphere. 

The only case in which the error of this assumption is appre- 
ciable will be considered in the Chapter on Eclipses. 

107. Tables of Refraction, — For the convenience of the reader 
who may wish to avail himself of the refraction tables without 
regard to the theory by which they are computed, I shall first 
explain the arrangement and use of those which are given at 
the end of this work. 

Since the amount of the refraction depends upon the density 
of the atmosphere, and this density varies with the pressure and 
the temperature, which arc indicated by the barometer and the 
thermometer, the tables give the refraction for a mean state of 
the atmosphere; and when the true refraction is required, supple- 
mentary tables are enii)loyed which give the correction of the 
mean rcfniction depending upon the observed height of the 
barometer and thermometer. 

Table I. srives the rcfniction when the barometer stands at 
30 inches and the thermometer (Fahrenheit's) at 50°. If we 

r = the refraction, 
z - the ai>paront zenith distance, 
Z -— the true zenith distance, 

: = ;: + r 

TVTiere great accuracy is not required, it suffices to take r 
directly from Table I. and to add it to z. (The resulting f is 
that zenith distance which we have heretofore denoted by f ' in 
the discussion of parallax.) The argument of this table is the 
apparent zenith distance z. 


Table IT. is Bessel's Refraction Table,* which is generally 
regarded as the most reliable of all the tables heretofore con- 
structed. In Column A of this table the refraction is regarded 
as a function of the apparent zenith distance z^ and the adopted 
form ofthis function is 

r= 0/5^7'^ tan z 

in which a varies slowly with the zeiyth distance, and its loga- 
rithm is therefore readily taken from the table with the argu- 
ment ^. The exponents A and X differ sensibly from unity only 
for great zenith distances, and also vary slowly; their values are 
therefore readily found from the table. 

The factor /9 depends upon the barometer. The actual pres- 
sure indicated by the barometer depends not only upon the 
height of the column, but also upon its temperature. It is, 
therefore, put under the form 

and log B and log 7^ arc given in the supplementary tables with 
the arguments "height of the barometer," and "height of the 
attached thermometer," respectively ; so that we have 

log /9 = log J? + log T 

Finally, log y is given directly in the supplementary table with 
the argument " external thermometer." This thermometer must 
be so exposed as to indicate truly the temperature of the atmo- 
sphere at the place of observation. 

In Column B of the table the refraction is regarded as a 
function of tlie true zenith distance ^ expressed under the form 

r = o'i9^>^'tan C 

and log a', A\ and /' are given in the table with the argument !^ ; 
/9 and x being found as before. 

Column A will be used when z is given to find ^ ; and Column 
B, when f is given to find z. 

Column C is intended for the computation of differential re- 
fraction, or the difference of refraction corresponding to small 

* From his AitronomUehe Untersuehungeny VoL I. 



difterences of zenith distance, and will be explained hereafter 
(Miorometric Observations, Vol. 11.). 

These tables extend only to 85° of zenith distance, bej'ond 
which no refraction table can be relied upon. There occur at 
times anomalous deviations of the refraction from the tabular 
value at all zenith distances; and these are most sensible at 
great zenith distances. Fortunately, almost all valuable astrono- 
mical observations can be made at zenith distances less than 
85°, and indeed less than 80° ; and within this last limit we 
are justified by experience in placing the greatest reliance in 
Bessel's Table. In an extreme case, where an observation is 
made within 5° of the horizon, we can compute an approximate 
value of the refraction by the aid of the following supplement- 
airy table, which is based upon actual observations made by 

A|»p. zon. 

log Refhict. 



85° (K 








































If we call R the refraction whose logarithm is given in this 

table, the refraction for a given state of the air will be found by 

the formula 

r = R?^r^ 

Example 1. — Given the apparent zenith distance z = 78° 
:50' 0", Barom. 20.770 inches. Attached Therm. — 0°.4 F., Ex- 
ternal Therm. — 2°.0 F. 

Wc find from Table H., Col. A, for 78° 30', 

logo 1 : 1.74981 

^l =r. 1.0032 

k = 1.0328 

and from the tables for barometer and thermometer, 

* Tabultt Regiomontantff p. 539. 


\ogB = + 0.00258 log r = + 0.04545 

log T= + 0.00127 

log /9 = + 0.00380 

Hence the refraction is computed as follows : 

log a = 1.74981 

A\ogfi = log iS^= + 0.00381 

>l log T' = log r^ = + 0.04694 

log tan z = 0.69154 

r = 310^.53 = 5' 10".53 log r = 2.49210 

The true zenith distance is, therefore, 78° 30' 0" + 5' 10".53 = 
78° 35' 10".53. 

Example 2. — Given the true zenith distance ^ = 78° 35' 
10".53, Barom. 29.770 inches, Attached Therm. — 0°.4 F., 
External Therm. — 2°.0 F. 

We find from Table H., Col. B, for 78° 35' 10", 

log a' = 1.74080 A' = 0.9967 A' = 1.0261 

and from the tables for barometer and thermometer, as before, 

log j5 = + 0.00253 log ^ = + 0.04545 

log T= + 0.00127 

log /5 = + 0.00380 

The refraction is then computed as follows : 

log a' = 1.74680 

A' log /9 = log /5^' = + 0.00379 

X' log r = log r^' = + 0.04663 

log tan : = 0.69489 

r = 310".53" = 5' 10".53 log r = 2.49211 

and the apparent zenith distance is therefore 78° 30'. 

Example 3. — Given z = 87° 30', barometer and thermometer 
as in the preceding examples. 

By the supplementary table above given, 

log E = 2.98269 

A = 1.0298 log /? = + 0.00380 log /3^ = + 0.00391 

;i = 1.2624 log r = + 0.04545 log r^ = + 0.05738 

r= 18'26".6 logr = 3.04398 


It is important in all cases where great precision is required 
that the barometer and thermometer be carefully verified, to sec 
that they give true indications. The zero points of thermo- 
meters are liable to change after a certain time, and inequalities 
in the bore of the tube are not uncommon. A special investi- 
gation of every thermometer is, therefore, necessary before it is 
applied in any delicate research. If the capillarity of the baro- 
meter has not been allowed for in adjusting the scale, it must be 
taken into account by the observer in each reading. 

We may obtain the true refraction for any state of the air 
within 1" or 2", very expeditiously, by taking the mean refrac- 
tion from Table L and correcting it by Table XIV. A, and Table 
XIV. B. The mode of using this table is obvious from its 
arrangement. Thus, in Example 1 we find 

from Table I., Mean refr. == 4' 38".9 

*< XIV. A, for Barom. 29.77, Corr. = — 2 . 
" XIV. B, « Therm. — 2^. " = + 32 . 

True rcfr. = 5' 9". 

which agrees with Bessel's value within l^.f). For greater 
accuracy, the height of the barometer should be reduced to tlie 
temperature 32° F., which is the standard assumed in these 
tables. The corrected height of the barometer in this example 
is 29.85, and the corresponding correction of the refraction 
would then be — 1"; consequently the true refraction would be 
5' 10", which is only 0".5 in error. 

These tables furnish good approximations even at great 
zenith distances. Thus, we find by them, in Example 3, r = 
18' 24". 

108. Investigation of tue refraction for.mula. — In tliis 
investigation we may, without sensible error, consider the earth 
as a sphere, and the atmosjihere as conii»osod of an infinite 
number of concentric si»hcrical strata, whose common centre is 
the centre of the earth, each of which is of unifonn density, and 
within which the path <»f a ray of light is a straight line. Let (' 
Fig. 16, be the rentre of the earth, A a point of obsen'ation on 
the surface; CA Z thv vertical line ; Aa\ a'b\ tV, &c. the vertical 
thicknesses of the con<*entric strata; Se a my of light from a star 
6', meeting the atmosphere at the point r, and suceessively re- 


fracted in the directions erf, rfe, &c. to the point A. The last 
direction of the ray is aA^ which, when the number of strata is 
supposed to be infinite, becomes a tangent to the curve ecA at -4, 
and consequently AaS' is the apparent direction of the star. Let 
the normals C5?, CI/, &c. be drawn to the successive strata. The 
angle Sef is the first angle of incidence, the angle Ced the firet 
angle of refraction. At any intermediate point between e and -4, 
as e, we have Ccrf, the supplement of the angle of incidence, and 
06, the angle of refraction. 
If now for any point, as c, in the path of the ray, we put 

i = the angle of incidence, 
/ = the angle of refraction, 

fi = the index of refraction for the stratum above c, 
ax' = " " " below c, 

then, Art. 105, 

^' = ^ (148) 

Bin/ fi 

If we put 

q = the normal Cc to the upper of the two strata, 
^ = '' Ch '' lower « « 

i' = the angle of incidence in the lower stratum, 
= 180<^ — Che, 

the rectilinear triangle Cbc gives 

sin i' q 

sin/ ^ 

which, with the above proportion, gives 

q fi sin i = q'f/ sin i' 

an equation which shows that the product of the normal to any 
stratum by its index of refraction and the sine of the angle of 
incidence is the same for any two consecutive strata; that is, it 
is a constant product for all the strata. If then we put 

z = the apparent zenith distance, 

a := the normal at the observer, or radius of the earth, 

/i^= the index of refraction of the air at the observer, 

we have, since z is the angle of incidence at the observer, 

qfi sin i = afi^ sin z (149) 


ill which the second member is constant for the same values of 
z and (JL^, 
JS'ow, we have from (148) 

tan J (i -/) = ^^1^ tan } 0' +/) 

But i — / is tlie refraction of the ray in passing from one stratum 
into the next ; and supposing, as we do, that the densities of the 
strata vary by infinitesimal increments, i — /is the differential of 
the refraction ; and we may, therefore, write J dr for tan i (t — f) 
and dii for fi! — ijl\ consequently, also, 2/i for /i' + /jl, and tan i for 
tan J (i +/) : hence we have 

dr = ^ tan i (150) 

which is the differential equation of the refraction. 

But, as both /i and i are variable, we cannot integrate this 
equation unless we can express i as a function of /i. Tliis 
we could do by means of (149) if the relation between q and 
[i were given, that is, if the law of the decrease of density of the 
air for increasing heights above the surface of the earth were 
known. This, however, is unknown, and we are obliged to 
make an hypothesis respecting this law, and ultimately to test 
the validity of the hypothesis by comparing the refractions com- 
puted by the resnlting formula with those obtained by direct 
obser\'ation. I shall first consider the hypothesis of IJougi'er, 
both on account of the simplicity of the resulting formula and 
of its historical interest.* 

109. First hypothesis, — Let it be assumed that the law of de- 
crease of density is such that some constant power of the refrac- 
tion index ft is recii>rocally i)roportional to the normal y, an 
hjliothesis expressed by the equation 

♦I shall consider but two hypotheses: the first, because it leads to the simple 
formula of BRAnLF.Y. which, though imperfect, is often useful a:» an approximate 
expression of the refraction; the second, because the tables fonne<l from it bv 
Bess EL have thus far appeared to be the most correct and in greatest acconlance with 
obserration, although on theoretical grounds even the hypothesis of Hkssel is open 
to objection. For a review of the labors of astronomers and physicists u|H)n this 
difficult subject, from the earliest times to the present, sec Pie Attronomitrh^ Struhhn* 
brerhuny in ihrer hiitoritchen Entwiekeluug (iar</f8tell(, von Db. C. Briiins. Leipiig, 



which with (149) gives 

sin i = / ^ jT sin zr (152) 

or, logarithmically, 

log sin I = n log fi + ^^Sy^j 

where the last term is constant. By differentiation, therefore. 

di dfi 

= n — 

tan i f^ 

which with (150) gives 

and, integrating, 

dr = ^ 

r = - +C 

To detennine the constant Q the integral is to be taken from 
the upper limit of the atmosphere to the surface of the earth. 
At the upper limit r = ; and if we put ^ = the value of i at that 
limit, we have 


At the lower limit the value of r is the whole atmospheric 
refraction, and i = z: hence 

r = - +0 

Eliminating the constant, we have 

r = '-^ (153) 

To find 1?, we have, by putting // = 1 in (152), since the density 
of the air at the upper limit is to be taken as zero, 

• ^ Dill S ,^ «. .^ 

sm »> = (154) 


Having then found n^ at the surface of the earth and suitably 


determined w, we find ?? by (154), and then r by (153). The two 
equations may be expressed in a single formula thus : 

which is knoANTi as Simpson's formula, but is in fact equivalent 
to the fonnula first given by Bouguer in 1729 in a memoir on 
refraction which gained the prize of the French Academy. 
From (154) we find 

sin 2 — sin t> /i^j* — 1 

sin 2 -f si*^ * /*o* + ^ 

tan i(^z — '»)= ^^—i:^ tan 1(2 + ^) 
and, reducing by (153), 

tan-r=:^^^^^ tan/r — -r\ (156) 

which is equivalent to Bradley's formula. If we are content to 
represent the refraction approximately by our fommla, we can 
write this in the form 

r = ^tan {z --ff) 

and we shall find, with Bradley, that for a mean state of the air 
corresi)onding to the barometer 20.G and thermometer 50° Fahr. 
we can express the observed refractions, very nearly, by taking 

g = 57".036, / =. 3. 

110. But, as we wish our formula to represent, if possible, the 
actual constitution of the atmosjihere, let us endeavor to test tlio 
hypothesis uiK>n which it rests. In order to correspond with tlie 
real state of nature, it is nccc:<sf(rt/ that the constitution of the atmo- 
sphere trhit'h the hj/pothesis inrohrs shouhl vot onh/ agree icith the 
ohserrnt refraction^ but also with the height of the barometerj and with 
(he ob^<ernfl (fiwination of heat as the altitude of the observer above (he 
earth's surface increases. 

The discussicm of the formula will be more simple if we sul>- 
stitute the density of the air in the place of the index of refrac- 
tion. Put 

\ -- the density of the air at the surface of the earth, 

— the density of the air at any point ahove the surface. 


The relation between d and /i, according to Optics, is expressed by 

Ai»— 1 = 4A^ (157) 

in which 4 & is a constant determined by experiment. Accord- 
ing to the experiments of Biot, 

4k = 0.000588768 

Since k is so small that its square will be inappreciable, we may 

fi = (l+ 4kd}i = l + 2kd (158) 

and, consequently, 

/i, = l + 2H 
A£o- = l + 2nH 

and (156) becomes, still neglecting A?, 

tan — r = nkd^ tan 
2 " 

i^-jr) (159) 

If we denote the horizontal refraction, or that for z = 90°, by r^ 
this formula gives 

tan — r« = nkd^ cot — r„ 


or tan -- r,, = ^/nkd^ 

and, puttmg the small arc — r^ for its tangent, 


"We can find d^ from the observed state of the barometer and 
thermometer at the surface of the earth, so that in order to com- 
pute the horizontal refraction by this formula, for the purpose 
of comparing it with the observed horizontal refraction, we have 
only to determine the value of n. 

X = the height of any assumed point in the atmosphere above 
the surface of the earth, 
d, py g = the density and pressure of the air, and the force of grav- 
ity, respectively, at that point, 
^0' Po9 9o = ^^® same quantities at the earth's surface. 


At an elevation greater than x by an infinitesimal distance d!r, 
the pressure jp is diminished by dp. The weight of a column of 
air whose height is dx^ density 5, and gravity g^ is expressed by 
gbdx^ and this is equal to the decrement of the pressure: hence 
the equation 

</p = — gbdx 

By the law of gravity, we have 

y = ^c 


(a + xy 
and hence 

dp = — gfi^d 

(a + xy 

Now, in the hj'pothesis under consideration, we have 

1 + 4A^\5^ 

a __l f^^'-^^^l l + ^kd V 
a + x^li^l ~\1 + Akdj 

or, neglecting the square of A, 


a-^- X 
which gives 

==l-2(n + l)A:a-0 


dp z=-.2g^a{ii + \)kddd 


p = g^a(ii^ l)AcJ« (162) 

no constant being necessarj-, since p and 8 vanish together. 
To compare this with the observed pressure i\^ let 

/ — - the height of a column of air of the density d^ which acted 
upon by the gravity g^ will be in equilibrium with the pres- 

in other words, let I be tlie hei<rht of a homogeneous atmosphere 
of the density 8^ which would exert the pressure p^. Tlien, by 
this definition, 

i>o = 9o\^ (163) 


which with (162) gives 

£=(« + l)f*^ (164) 

At the surface of the earth, where p becomes p^ and d becomes 
d^ this equation gives 

l = (n+l)^.A:^, (165) 

whence i 

and this reduces the expression of the horizontal refraction (160) to 

Taking as the unit of density the value of d^ which corre- 
sponds to the barometer 0.76 metres and thermometer 0° C, 
we have, according to BiOT, 

4A^o = ^-^00588768 

The constant I for this state of the air is the height of a homo- 
geneous atmosphere which would produce the pressure 0"'.76 of 
the barometer when the temperature is 0° C. ; and this height is 
to that of the barometric column as the density of mercury is to 
that of the air. According to Regnault, for Barom. 0"*.76 and 
Therm. 0° C, mercury is 10517.3 times as heavy as air: hence 

we have 

I = 0« 76 X 10517.3 = 7993-.15 

For a we shall here use the mean radius of the earth, since we 
have supposed the earth to be spherical, or 

a = 6366738 metres 
which gives 

- = 0.00125545 (167) 

Substituting these values in (166), we find, after dividing by 
sin V to reduce to seconds, 

r^ = 1824" = 30' 24" 

. But, according to Arqelander*s observations, we should have 


for Barom. O-^.TG and Therm. 0^ C, r^= 37' 31''; and the hypothesis 
therefore gives the horizontal refraction too small by more than 7'. 

111. The hypothesis can be tested further by examining 
whether it represents the law of decreasing temperatures for 
increasing heights in the atmosphere. In the first place, we 
observe that m this hypothesis the densities of the strata of the atmo- 
sphere decrease in arithmetical j)rogression ichen the altitudes increase 
in arithmetical progression. For, since x is very small in compari- 
son with a, we have very nearly 

a - X 

a + X a 

and hence 

f = 2(n + l)H(l-^) 

or, by (165), 


= 2i(l-^) (168) 

which shows that equal increments of x correspond to equal 
decrements of d. 

This last equation also gives for the upper limit of the atmo- 
sphere, where d = 0,x-=2l; that is, in this hypothesis the height of 
(he atmosphere is double that of a homogeneous atmosphere of tlic same 

Again, we have, by (1G4), (105), and (168), 

?Ao^i==l_^ (169) 

p,d d, 21 ^ ^ 

The function ^ expresses the law of heat of the strata of the 

atmosphere. For let r^ be the temperature at the surface of the 

earth, r the temperature at the height x. If the temperature 

were r^ in both cases, we should have 

^ = - (170) 

Po ^0 

but when the temperature is changed from r^ to r the density is 
diminished in the ratio 1 + « (r — - r,,) : 1, t being a constant which^ 


is known from experiment; so that the true relation between 
the pressures and densities at different temperatures is expressed 
by the known formula 

Po % 


^ = l + e(r-ro) (171) 

which combined with (169) gives 

and hence equal increments of z correspond to equal decrements 
of r. Hence, m this hypothesis^ the heat of the strata of the atmo- 
sphere decreases as their density in arithmetical progression. The 
value of e, according to Rudbero and Regnault, is very nearly 

1 21 

— . Hence we must ascend to a height ^ = 58.6 metres, in 

order to experience a decrease of temperature of 1° C. But, 
according to the observations of Gay Lussac in his celebrated 
balloon ascension at Paris (in the year 1804), the decrease of 
temperature was 40°.25 C. for a height of 6980 metres, or 1° C. 
for 173 metres, so that in the hypothesis under consideration 
the height is altogether too small, or the decrease of temperature 
is too rapid. This hypothesis, therefore, is not sustained either 
by the observed refraction or by the observed law of the decrease 
of temperature. 

112. Second hypothesis. — ^Before proposing a new hypothesis, 
let us determine the relation between the height and the density 
of the air at that height, when the atmosphere is assumed to be 
throughout of the same temperature, in which case we should 
have the condition (170). Resuming the differential equation 

\a + xf 

=1 —3 

a -\- X 


in which 5 is a new variable verj- nearly proportional to z. We 
then have 

dp = — g^fidds 

which with the supposition (170) gives 

^/> _ _ 90^ Ms 

P Po 


in which the logarithm is Napierian. The constant being 
determined so that p becomes p^ when 5 = 0, w^e have 

and therefore 

1 P ^0^0 ^* 

log f = — ^ ^5 = — -r- 

Po Po 

where I has the value (163). Ilcnce, e being the Napierian base, 

I = J = e-T (172) 

Po \ 

which is the expression of the law of decreasing densities upon 
the supposition of a uniform temperature. In our first hypo- 
thesis the temperatures decrease, but nevertheless too rapidly. 
We nuisiy thm^ frame an hypothesis between thai and the hypothesis of 
a uniform temperature. 

Now, in our first hj-pothcsis we have by (169), >vithin terms 
involving the second and higher powers of 5, 

p^o ^ as^ 

and in the hypothesis of a uniform temperature, 

^ = 1 


The arithmetical mean between these would be 

P^9 _^^ ^ 


buty as we have no reason for assuming exactly the arithmetical 
mean, Bessel proposes to take 

0=e * = l_- + -(-)-&c. (178) 

h being a new constant to be determined so as to satisfy the observed 
refractions. This equation, which we shall adopt as our second 
hypothesis, expresses the assumed law of decreasing tempe- 
ratures, since, by (171), it amounts to assuming 


l + f(r~r,) = c-* (174) 

and it follows that in this hypothesis the temperatures will not 
decrease in arithmetical progression with increasing heights, 
though they will do so very nearly for the smaller values of 5, 
that is, near the earth's surface. 
Now, combining the supposition (173) with the equation 

dp = — g^dds 

we have 

dp gji^ « , CL ^ ^ 
-£. = -^ils:ehds = eh ds 

P Po I 

Integrating and determining the constant so that for s = Oyp 
becomes p^, we have 

which with (173) gives* 

3 = 3.e-f(«*-l)+? 

Inasmuch as the law of the densities thus expressed is still 
hypothetical, we may simplify the exponent of e. For if h is 
much greater than I (as is afterwards shown), we may in this ex- 

" as 

ponent put 6 * — ^ "^ T *^^ ^^ shsll have as the expression 
of our hypothesis 

d = d^e l"^A=^o« * « ^^ ' ^^ 

* B188BL. Fundamenta ABtronomiSy p. 2S. 
Vol. L— 10 


By comparing this with (172), we see that this new hypothesis 
differs from that of a uniform temperature by the correction r- 

applied to the exponent of e. 
Putting, for brevity, 

« ^ Id /'■"•An 

we have 

d = ^ c-^ (177) 

in which /9 is constant. Tliis expression of the density is to be 
introduced into tlie differential equation of the refraction (150). 
Now, by (149), in which j = a + x, we liavo 

. . aji^ sin^r (1 — s) pt^ sin-? 
sin 1 =: — ^^ / '-^ 

(a + x)ii 

f ATI 1 



iin t 

_ (l-a) 

sin z 

toil • 


— sin' I) 


- «)* sin^ 



(1 — «) sin z 


r>R* * 1 

1 "' \a. (1. 

«*^ Bin* ; 


From the equation /i* = 1 + 4 W we deduce 

and if we introduce as a constant tlic quantity 

1 + \k\ 
(which for Barom. O-.TG and Tlicrm. 0° C. is a = 0.000294211) 

a — 



We might neglect the square of A*, and consequently, also, that of 


a, with hardly appreciable error, and then this expression would 
become simply a — , but for greater accuracy we can retain the 

denominator, employing its mean value, as it varies within very 
narrow limits. For its greatest value, when 8 = 8^, is = 1, 
and its least value, when 5 = 0, is = 1 — 2a, and the mean 
between these values is 1 — a. Hence we shall take 

d/i a dd 

M 1 — a ^0 

In the denominator of the value of tan { we have also to sub- 

l_^=:l-_^ + ^^^ 

,i\ 1 + m, 

Therefore, substituting in (150), we have 

a Bin Z (1 — 5) — 

dr= ^' 


(1 — o) [coB» 2 — 2o/ 1 — ~ \ + (25 — 5») sin» z] * 

or, by (177), 

, — o/g sin 2 (l-^s) e^^'ds 

(1 — a) [co8»2r — 2o (1 — «-^) + (2s — s*) sin»2] i 

In the integration of this equation we may change the sign of 
the second member, since our object is only to obtain the 
numerical value of r. It is apparent that if we put 1 for 1 — 5 
in the numerator of this expression, and also neglect the term 
5* sin* 2 in the denominator, the error will be almost or quite 
insensible ; but, not to reject terms without examination, let us 
develop the expression into series. For this purpose, put the 

radical in the denominator under the form i/ y^ — ^ sin* Zy in 

y = [cos» 2r — 2a (1 — e"^ + 25 sin* z^^ 

1—5 _ 1 — 5 /^ __5*sin»2r\-l 

(y* — 5* sin* 2:) i y \ y' / 

1 — 5 / ^ .5' sin* z 


1 2sv' — s*Bin'2 . 

= ^ ^ &c. 

y 22/' 

Hence, restoring the value of y, we have 

a,p sin z e'^^ds 

dr = 

(1 — a) [cos' 2r — 2o (1 — c-^) + 28 8in» z]i 

ap sin z e^^sds [cos* z — 2o (1 — e"^*) + 1 5 sin* -?] 
(1 — a) [cos« z — 2a(l — e-^)+ 2s sin* ^]l 

— &c (179) 

We shall hereafter show that the second term of this develop- 
ment is insensible. Confining ourselves for the present to the 
first term, let us, after the method of Laplace, introduce the new 
variable s' such that 

8 = ^ + '^(l^f^ (180) 

Sin* 2 

tlicn this term takes the form 

dr = o^Bi"^^"^^^^ (181) 

(1 — o) [cos* z + 25' sin* 2] 1 ^ ' 

in which we have yet to reduce the numerator to a function of 
the new variable s'. Now, by Lagrange's Theorem* when 

* See Peirce's Curres and Fanctions, Vol. I. Art. 181. For the conyenience of 
the reader, howeyer, I add the following demonstration of this theorem. It is pro- 
posed to develop the function u :=fy in a series of aacending powers of x, z andjr 
being connected by the equation 

and the functions/and ^ being given. If from this equation y could be found at an 
explicit function of x and substituted in the equation ti =:/y, the deyelopment could 
be effected at once by Maclaurin's Theorem, according to which we should haye 



« = Wo + D,u^ 4- /),»Mo —-+.... + ^,"Wo-— h &c. 

l.z l.z....fl 

where Ug, D^u^ &c. denote the values of u and its successive derivatives when z = 0. 

It is proposed to find the values of the derivatives without recourse to the elimination 

of y, as this elimination is often impracticable. For brevity, put 1' = ^; then the 

derivatives of 

y = t^xT 

relatively to x and / are 


8 = Sf -\- af8 

we have 

fs =/5' + -f [f^. 2)/^' ] + ^ 2) [(fO'.D/^'] 

+ f;|;3-Z>'[(K)'.i>/y]+&c. 

in which/ and y> denote any functions whatever, and 2), 2)^, &c. 
the successive derivatives of the functions to which they are 
prefixed. Hence, by putting 

1 — e"^' 

fs = e''^' fs = — —, 

Bin' J? 

this theorem gives 

2>.y = r+ xDJD^y l>,y = 1 + z B^YD.y 

whence, eliminating z, 

Multiplying this by Dji^ it giyes 

Z>, « = TD^u (a) 

The deriyatiye of this equation relatively to < is 

This is a general theorem, whateyer function ti is of y, and consequently, also, what- 
eyer ftinction D^u is of y. We may then substitute in it the function Y^D^u for D^u^ 
and we shall haye 

Z>,[ Y^D^u\ = Z>, [ F» + i/),t«] (6) 

Now, the successiye deriyatives of (a) relatiyely to z are, by the suecessiye appli- 
cation of (6), making n = 1, 2, 8, &o.. 

But when z == 0, we haye y = t, F = ^^, and hence 

tfo=A />,iio = ^.2>A ... />.''tio = Z>«-i[(0O"^/O 

where the subscript letter of the D is omitted in the second members as unnecessary^ 
since t is now the only yariable. These yalues substituted in Maclaurin's Theorem 
giye Lagrange's Theorem : 

150 BB7BA0TI0K. 

e-P'=e-f'' ^ ri - e-^n «-«"' 

fov? z *- J 

- , o''^ ^ [(1 - «-"•')*« -"'1 

'*''—- ■D' [(1 - e -'•')* «-"•'] 

1.2.3 8in«z 

**'^ i>-*[(l-c-^'')«(j-^''] 

1. 2.3. ..nsin**;? 
— &c. (182) 

But we have in the numerator of (181) 

and hence, differentiating (182) and substituting the result in 
(181), we find 

dr= -I^_zdl .L-.^.'+_^2)^(l-e-^•')e-^n 

(1 — o) [co8« 2 + 2 8' 8in« 2r]M ^sin'^r ^^ ^ J 


+rw-.^[(i-''-''")' *'-""] 

+ 1.2.3.. .nBin»>. -^[(^--~^*>-"'*1 

+ &c. I (188) 

To effect the differentiations expressed in the several terms of 
this series, we take tlic general exi)re88ion 

(l_e-^'')"e-^'' = (-c-^'' + l)*c-^'' 

where the upper sign is to be used when n is even, and the lower 
sign when n is odd. Differentiating this n times suceessivelyi 
we have 

i>[(l— c-^'7e-^'']=+/?»[(n+l)»c-<* + *>^''— n».n«— ^''+40.] 


by means of which, making n = 1 . 2 . 3 . . . successively, we 
reduce (183) to the following form : 

(1 — a)[C08»2+2s'8in»2]il Bin«2^ ' 

^1.2 8in*^^ ^ ' 



4- &c. > 

We have now to integrate the terms of this series, after having 
multiplied eaeh by the factor without the brackets. The inte- 
grals are to be taken from the surface of the earth, where 5 = 0, 
to the upper limit of the atmosphere ; that is, q being the nor- 
mal to any stratum (Art. 108), they are to be taken between the 
limits 5 = a and q = a + Sy H being the height of the atmo- 
sphere. Now, this height is not known ; but since at the upper 
limit the density is zero and beyond this limit the ray suffers 
no refraction to infinity, we can without error take the integrals 
between the limits q = a and j = oo , i.e. between 5 = and 
5 = 1. But we may make the upper limit of s also equal to in- 
finity. For, by (176), ^ will not differ greatly from ^, and conse- 
quently will be a very large number, nearly equal to 800, as we 
find from (167) ; hence for 5 = 1 we have in (172) d = ^" 

(2.718 . .)~ 

which will be sensibly equal to zero, and consequently the same 
as we should find by taking 5 = oo . Hence the integrals may 
be taken between the limits 5 = and 5 = oo ; consequently, 
also, according to (180), between the limits 5' = and 5' = oo . 
Now, as every term of the series will be of the form 


[co8» z + 2sf sin' 2:]* [cot' z + 28r\i 

multiplied by constants, we have only to integrate this general 
form. Let t be a new variable, such that 

cot»2f + 25' = ^ (186) 


then (185) becomes 

If n 

the integral of which is to be taken from t = 

\^ooiz=T (187) 

to < = 00 , which are the limits given by (186) for s^=0 and 
a' = 00 . If, therefore, we denote by '4/ {n) a function such that 


4(7i) = <j^r°'(ttc-" (188) 

the integral of (185) will become 

•^0 [cos»2+2s'8in«2:]* Vn 

Substituting this value in (184), making successively n = 1, 2, 8, 
&c., we find the following expression of the refraction: 



+ 15^ pVs) -Kl)] 


+ 1.2Tsin'g E-^^+W-^*- 3 + (3) + 2*.34(2)-4(l)] 
+ &C. I (190) 

which, since wc have in general 

1^1.2 1.2.3 + * 


can also be written as follows :"*" 

e •!«•* 4 (1) 

, ' sin*2r ^^ '^ 

•1/2^ / • 

.+ &C. 

113. The only remaining difficulty is to determine the func- 
tion i^n), (188). In the case of the horizontal refraction, where 
cot ^ = and therefore also r= 0, this function becomes 
independent of (n), and reduces to the well-known integralf 


~ -« v^ 

dte-'' = -^-^ (192) 


* Laplace, Mieanique Ciletie, Vol. IV. p. 186 (Bowditch's Translation) ; where, 
however, - stands in the place of the more general symbol p here employed. This 

form of the refraction is due to E&amp, Analyse det rifraetiona tutronomiquea et ter- 
rettret, Strasbourg, 1799. 
f This useful definite integral may be readily obtained as foUows. Put k = 

dte — tt. Then, since the definite integral is independent of the Tariable, we 

doe ^^ , and, multiplying these expressions together, 

the order of integration being arbitrary. Let 

V z=iu; whence dv = idu 
(for in integrating, regarding v as yariable, t is regarded as constant) : then we have 
*»=r*r*<fM. A. <«-«a + «!«)= f^duf^dt, te-t^(^-^w») 

^f^du.-—! -=}(tan-»oo — tan-»0)=^ 

•^0 2(l + tt») ^^ -^ 4 


•^0 ~" 2 


where tt = 3.1415926 .... The expression for the horizontal 
refraction is therefore found at once by putting ^i/n for '^^ (w) 
in every term of (191), and sin z = lj namely: 

^ 1.2.3 

+ &c. 

For small values of T, that is, for great zenith distances, we 
may obtain the value of the integral in (188) by a series of 
ascending powers of T. We have 

I dte-^^={ dte-^^—) dte-^^ (194) 

The first integral of the second member is given by (192). The 
second is 

fdte -''= C''dtll-t'+-^ ?!_+&c.\ 

^0 Jo \ ^ 1.2 1.2.3 ^ / 

rpi 1 /TT6 1 rpt 

= T—— + -^'- ^— . — + &c. (195) 

3 1.2 5 1.2.3 7 ^ 

Another development for the same case is obtained by the suc- 
cessive application of the method of integration by parts, as 

Jdte-^^ = t e-" + 2ft^dte-^^ 

9 9a /• 

. t*dte-" 

♦ By the formula yxt/v - ary ^fydz^ making always x = ? ". and dy succes- 
Bively = di, Odt, t\dt, &c. 


whence, by introducing the limits, 

•^0 \ ^ 3 ^ 3.5 ^3.5.7^ / ^ ^ 

As the denominators increase, these series finally become con- 
vergent for all values of T; but they are convenient only for 
small values. 

For the greater values of Tj a development according to the 
descending powers may be obtained, also by the method of 
integration by parts, as follows :* We have 


^ T 2T\ 2T«(2 T«)« (2 T*)» 

^ 1.3.5...(2n-l) l 1.3.5...(2n + l) r°- dt ^,» 
~ (2 T*)* J"*" 2*+* ^T <*"+» 

The sum of a number of consecutive terms of this series is 
alternately greater and less than the value of the integral. But 
since the factors of the numerators increase, the series will at 
last become divergent for any value of T. Nevertheless, if we 
stop at any term, the sum of all the remaiimg terms will be less than 
this term; for if we take the sum of all the terms in the brackets, 
the sum of the remaining terms is 

_ 1.3.5...(2n + l) r" dt _„ 
2» + * •^ T <"•+* 

* By the formulay*z dy = xy — /y dz, making always dy = t dt e " , and x 
BnceessiYelv = -, -.» 7:1 &c. 


The integral in this expression is evidently less than the product 
of the integral 




multiplied by the greatest value of e-" between the limits Tand 
00 , and this greatest value is e~^. Hence the above remainder 
is always numerically less than 

1.3.5....(2n— 1) -rr 

which expression is nothing more than the last term of the series 
(when multiplied by the factor without the brackets), taken with 
a contrary sign. Hence, if we do not continue the summation 
until the terms begin to increase, but stop at some sufficiently 
small term, the error of the result wall always be less than this 

Finally, the integral may be developed in the form of a con- 
tinued fraction, as was shown by Laplace. Putting for brevity 

4(n) = Mo = — (l ^,+ -^-^-^r^ + &c.\ (198) 

and denoting the successive derivatives of Uq relatively to Tby 
u^y 2^2, &c., we have first 

u, = -—+ -— - -~ + &c. (199) 


u, = 2Tii, — l (200) 

Differentiating this equation, successively, we have 

t/, = 2Tt/j + 2t/o 

W3 = 2rw, + 4i/i 

u^ = 2Tu^ + 6ti, 


or, in general, 

t/n + i =2Tu^ + 2nUn^i 

n having any value in the series 1 . 2 . 3 . 4 . . . &c. 


Hence we derive 

Un 2n 

or, putting 

* = ^ (201) 




Mn-l 1 /*\J«, + i 

By (200) we have 


«o = 

2r— ^ 


But £rom (202), by making n suocessirely 1, 2, 8, &c., we have 


(r .. '■'& 

which successively substituted in (203) give 

22V. = -i^ 


1 + 

1 + 2A 

1 + ?* 


1 + &c. (204) 

This can be employed for all values of ST, but when k exceeds J 
it will be more convenient to employ (195) or (196). 
The successive approximating fractions of (204) are 

1 1 l + 2k 1 + bk 1 + 9A + 8A« 

' z —y ^.J 1 — > &c. 

1 1 + /: IJ^U 1 + 6A: + 3Af« 1 + lOit + 15A» 
and, in general, denoting the n** approximating fraction by — , 


fln an-^i + (n — 1) kan-2 
bn "" 6„ -1 + (n — 1) kbn^2 

By the preceding methods, then, the function '^n) can be 
computed for any value of Jl A table containing the logarithm 
of this function for all values of J* from to 10, is given by 
Bessel {Fundamenta Astronomies^ pp. 36, 37), being an extension 
of that first constructed by Kramp. By the aid of this table the 
computation of the refraction is greatly facilitated. 

114. Let us now examine the second term of (179.) This term 
wdll have its greatest value in the horizontal refraction, when 
z = 90°, in which case it reduces to 

ape-^ Sds [I 5 — 2a (1 — «-^)] 
(1 — a) [25 — 2a (1 — e-^O]* 

Moreover, the most sensible part of the integral corresponds to 
small values of 5, and thereforq, since a is also very small, we 
may put 2a (1 — e*"^') = 2a^s. The integral thus becomes 

2i(l_a)(l —afiy^O 

Now we have, by integrating by parts, 

J /9 ' 2/9 •^ 

and hence 


Putting fis = 7?^ this becomes, by (192), 

Hence the term becomes 

g (3 — 4a/9) \x_ 

8(1— a)(l — a;9)l \2/9 


Taking Bessel's value of A = 116865.8 toises* = 227775.7 
metres, and the value of I = 7993.15 metres (p. 141), we find by 
(176) ^ = 768.57. Substituting this and a = 0.000294211 (p. 146), 
the value of the above expression, reduced to seconds of arc by 
dividing by sin V\ is found to be only 0'^72, which in the hori- 
zontal refraction is insignificant This term, therefore, can be 
neglected (and consequently also all the subsequent terms), and 
the formula (191) may be regarded as the rigorous expression of 
the refraction. 

115. In order to compute the refraction by (191), it only re- 
mains to determine the constants a and ^. The constant a 
might be found from (178) by employing the value of k deter- 
mined by BiOT by direct experiment upon the refractive power 
of atmospheric air, but in order that the formula may represent 
as nearly as possible the observed refractions, Bessel preferred 
to determine both a and ^ from observations.f 

Now, a depends upon the density of the air at the place of 
observation, and is, therefore, a function of the pressure and 
temperature; and ^, which involves i, also depends upon the ther- 
mometer, since by the definition of I it must vary with the tem- 
perature. The constants must, then, be determined for some 
assumed normal state of the air, and we must have the means 
of deducing their values for any other given state. Let 

p^ = the assumod normal pressure, 

T^ = " " tomporaturo, 

p =z the observed pressure, 

T = " " temperature, 

d^ = the normal density corresponding to p^ and t©, 

d = the density corresponding top and t; 

* Fundamenta Aitronomia, p. 40. 

f It should be obserred that the assumed expression of the density (177) may 
represent yarious hypotheses, according to the form given to j9. Thus, if we put 

fi = If ^^ ^^® *^® ^<>™* (1"2) which expresses the hypothesis of a uniform tem- 
perature. We may therefore readily examine how far that hypothesis is in error in 
the horizontal refraction; for by taking the reciprocal of (167) we have in this case 
3 = 796.53, and hence with a — 0.00029411 we find, by taking fifteen terms of the 
series (193). r^ = 39* 64''.6, which corresponds to Barom. 0^. 76, and Therm. 0*» C. 
This 18 2' 28".5 greater than the value given by Aboelamdicr's Observations (p. 141). 
Our first hypothesis gave a result too small by more than 7', and hence a true hypo- 
thesis must be intermediate between these, as we have already shown Arom a con* 


then we have by (171) 

d = , 5> ,.L 

in which « is the coefficient of expansion of atmospheric air, or 
the expansion for 1° of the thermometer. If the thermometer is 
Ceniigradej we have, according to Bessel,* 

e = 0.0036438 

From (178) it follows that a is sensibly proportional to the 
density, and hence if we put 

a^ = the value of a for the normal density 8^ 
we have, for any given state of the air, 


l + ^(^ — '^o) Po 


in which for p and Pq we may use the heights of the barometric 
column, provided these heights are reduced to the same tem- 
perature of the mercury and of the scales. 
Again, if 

l^ = the height of a homogeneous atmosphere of the temperatoro 
Tq and any given pressure and density, 

then the height I for the same pressure and density, when the 
temperature is r, is 

Z = ?,[l + e(T~r„)] (206) 

The normal state of the air adopted by Bessel in the determi- 
nation of the constants, so as to represent Bradley*s obser\'a- 
tions, made at the Greenwich Observatory in the years 1750- 
1762, was a mean state corresponding to the barometer 29.6 
inches, and thermometer 50° Fahrenheit = 10° Centigrade; and 
for this state he found 

a„ = 0.000278953 

sideration of the law of temperatures. At the same time, we see that the hjpothetif 
of a uniform temperature is nearer to the truth than the first hypothesis, and we are 
80 far justified in adhering to the form 6 = d^^-fi* with the modification of substi- 
tuting a corrected value of 3. 

* This Talue, determined by Bessel, from the obserrations of stars, differs slightlj 
from the value ^fy more recently determined by Kudbkbq and Rkonault hj diitoi 
experiments upon the refractive power of the air. 


or, dividing by sin 1", 

ao = 57" .538 


h = 116865.8 toises = 227775.7 motros. 

For the constant l^ at the normal temperature 50° F., Bessel 


l^ = 4226.05 toises = 8236.73 metres.* 

Since the strata of the atmosphere are supposed to be parallel to 
the earth's surface, Bessel employed for a the radius of curva- 
ture of the meridian for the latitude of Greenwich (the observa- 
tions of Bradley being taken in the meridian), and, in accordance 
with the compression of the earth assumed at the time when 
this investigation was made, he took 

a = 6372970 metres. 
Uence we have 

A = ^-^" • - = 745.747 

These values of a^, and ^„ being substituted for a and ^ in 
(193), the horizontal refraction is found to be only about V too 
great, which is hardly greater than the probable error of the 
obser\^ed horizontal refraction. At zenith distances less than 
85°, however, Bessel afterw-ards found that the refraction com- 
puted with these values of the constants required to be multi- 
plied by the factor 1.003282 in order to represent the Konigsberg 

116. By the preceding formulfe, then, the values of the con- 
stants a and /9 can be found for any state of the air, as given by 
the barometer and thermometer at the place of observation, and 
then the true refraction might be directly computed by (191). 
But, as this computation would be too troublesome in practice, 
the mean refraction is computed for the as^med normal values 
of a and ^, and given in the refraction tables. From this mean 

* According to the later determination of Regnault which we hare used on p. 143, 
we should have /o=S2S6.1 metres. The difference does not affect the value of 
Bbssbl's tables, which are con3tructed to represent actual obserrations. 
Vol. L— 11 


refraction we must deduce the true refraction in any case by 
applying proper corrections depending upon the observed state 
of the barometer and thermometer. For &cility of logarithmic 
computation, Bessel adopted the form 

r = r.ltYi "^ Y (207) 

in which r, is the tabular refraction corresponding to p^ and r^ 
and r is the refraction corresponding to the observed p and r. 
Let us see what interpretation must be given to the exponents 
A and X. If the pressure remained p^ the refraction correspond- 
ing to the temperature r would be 

, dr . ^ , rfV (r — T.)« , . 

or, with sofficient precision, 



In like manner, if the temperature were constant, and the pres- 
sure is increased by the quantity p — p^ the refraction would 
become nearly 

Ilcncc, when both pressure and temperature vary, we shall have, 
ver}' nearly, 

Xow, putting — in (207) under the form 1 + — --^,anddevelop- 

Po Po 

ing by the binomial theorem, we have 

r = r, 1 1 + - (;> -;>o) + &c. | X { 1 -^e (r-rj + 4e. | 
Therefore, neglecting the smaller terms, we must have 

A = ^.'Ji ^ = --T (209) 


to determine which we are now to find the derivatives of (191) 
relatively to jp and r. Put 

X = -^ (210) 

Bin* z ^ ^ 

and ?! = 1^ (1), Ja = 2^(2), g, = 3^(3), &c., or in general 

ff. = n'*« 4(n) (211) 

then, if we also put 

q = xe-^q,^^e-^-q, + ^ ^ "^ ^ ^-""g^H- &c. (212) 

the formula (191) becomes 

(l-a)r=Bin«2^.e (213) 

in which, since the variations of ^— — in (191) are sensibly the 

same as those of a, we may regard 1 — a as constant. Differen- 
tiating this, observing that Q varies with both j? and r, while ^ 
varies only with r, we have 

(1 — o) — = sm* z \ -^ 

^ ^ dp \fi dp 

) (214) 
dr \^ dr "^ "^2^ "d^ 

^ dr \ R d-r ^ ^9 A 

In differentiating §, it will be convenient to regard it as a func- 
tion of the two variables x and ^, the quantities y^ jj, &c. vary- 
ing only with ^. We have, since ^ does not vary with p, 

dQ^dQ dx ^215) 

dp dx dp 

and since both x and Q vary with r, 
From (212) we find 


in which 

«' = ^^"' ffi + j^ e"'' 2q, + ^-^ 6-3. 3j^ + &c. (218) 


^ = a:e-ii' + -^e-^^' + &c. (219) 

d/9 <f/9 ^1.2 da ^ ^ ^ 

in which we have generally, by (211), 

But by (200), in which ?«, = i//(n), we have 
and by (187) 

dT -^*- 


cot* 2 cot Z 

= ng — 

2 i/2^" 



Snbstitntinn^ the values of this expression for ?i = 1, 2, 3, &c. in 
(219), we have 

The first series in this expression — Q^. The second, when 
e~'j t' " '^, &c. are developed in series, becomes 


x -!- x' 4- a:* -f &c. = 

J "^ I 



and hence 

§ = ?2^1q'^J2^.^ (220) 

We have, further, from (210) and the values of a, I, and ^ in the 
preceding article, 

dx X da X a X 
dp a dp~~ a p """^ 

dfi d? dl a 

dr ~~ dl dr" Z« ' ^ ^ ~" 

h — l 


dx X da X dp 2h — I 

dr a dr ' p dr h — l 

Substituting these values in (215) and (216), and then substituting 
in (214), we find* 

.^ .dr . - /2 ^, \—x 

These formulse are to be computed with the normal values of a, 
^, r, I, and p^ and for the different zenith distances, after which 
A and X are computed by (209). The values of A and 3i thus 
found are given in Table 11. 

117. Finally, in tabulating the formula (207), Bessel puts 

r^ = a tan z (222) 


= A r = 

l + '-(^~^o) 

(where a and )9 no longer have the same signification as in the 
preceding articles). 

^—~- — ■ * — 

* Bkssel, Fundamenta Attronomia^ p. 34. 


The true refraction then takes the form 

r = aP^r^ tan z (223) 

The quantity here denoted by /J is the ratio of the observed and 
normal heights of the barometer, both being reduced to the same 
temperature of the mercury and of their scales. First, to correct 
for the temperature of the scale, let b^^y b^^\ or 6^"*> denote the ob- 
served reading of the barometer scale according as it is graduated 
in Paris lines, English inches, or French metres. The standard 
temperatures of the Paris line is 13° Reaumur, of the English inch 
62° Fahrenheit, and of the French metre 0° Centigrade ; that is, 
the graduations of the several scales indicate tnie heights only 
when the attached thermometers indicate these temperatures 
respectively. The expansion of brass from the freezing point to 
the boiling point is .0018782 of its length at the freezing point 
If then the reading of the attached thermometer is denoted 
either by r',/', or c', according as it is Keaumur's, Fahrenheit's, 
or the Centigrade, the true height observ^ed will be (putting s = 

l + -r' 1+— (/' — 32) i + JL.c' 

6(0. '' , 6(->> ''' \ 60^). ;^ 

1 + -.13 l^_-^.30 ^ 

'so 180 


80+^ J,, 180_+_(r^J2)£ ^(.) 100 + c-. 

80 + 135' 180 + 306* 100 ^ ^ 

where the multipliers 1 + — r', &c. evidently reduce the reading 

to what it would have been if the observed temperature had been 

that of freezing, and the divisors 1 + — • 13, &c. further reduce 

those to the respective temperatures of graduation, and conse- 
quently give the true heights. 

This true height of the mercury will be proportional to tlie 
j)ressure only when the temperature of the mercury is constant. 
AVo must, therefore, reduce the height to what it would be if the 
temperature were equal to the adopted normal temperature, wliich 
is iu our table 8° Keaumur ^-^ 50° F. =^ 10° C. Now, mercury 

expands --- of its volume at the freezing point of water, when 

*' 65.6 


its temperature is raised from that point to the boiling point of 

water. Hence, putting q = — , the above heights will be reduced 

to the normal temperature by multiplying them respectively by 
the factors 

SO+Sq 180 + 18g 100 + lOg ,^26^ 

80 + rV l80+(/' — 32)j' 100 + c'j ^" ^ 

The normal height of the barometer adopted by Bessel was 29.6 
inches of Bradley's instrument, or 333.28 Paris lines ; but it after- 
wards appeared that this instrument gave the heights too small 
by J a Paris line, so that the normal height in the tables is 333.78 
Paris lines, at the adopted normal temperature of 8° R. Reducing 
this to the standard temperature of the Paris line = 13° R., we 

In comparing this with the observed heights, the 6^*^ and 6^"*^ must 
be reduced to lines by observing that one English inch = 11.2595 
Paris lines, and one metre = 443.296 Paris lines. Making this 

reduction, the value of j(9 = - is found by dividing the product 

of (224) and (225) by (226). The result may then be separated 
into two factors, one of which depends upon the observed height 
of the barometric column, and the other upon the attached ther- 
mometer ; so that if we put 

jB = 

^»('> SO + Sq 

333.78 80 + 85 

— h(e) 11-25^^ 80 + 13^ 180 + 18g 
"~ ' 333.78 ' 80 -f~87 ' 180 + 30s 

— 5(«) ^^-^^^ 80 + Us 100 + lOg ^ (^^^^ 
■" * 333.78 ' 80 4- 85 ' 100 



_ 80 + r's _ 180 + (f — 32)s _ 100 + c's 
"~ 80 + r*q ~" 180 -}-(/' — 32) ^ ~ 100 + c'q 

we shall have ^ = BTj or 

log /9 = log iJ + log T (228) 


The quantity y would be computed directly under the form 

r = 

1 + Kt-t,) 

if To were at once the freezing point and the normal temperature 
of the tables ; for e is properly the expansion of the air for each 
degree of the thennometer above the freezing point, the density 
of the air at this point being taken as the unit of density. But 
if the normal temperature is denoted by r^,, that of the freezing 
point by r^, the observed by r, we shall have 

l+c(r, -rO 

an expression which, if we neglect the square of e, will be reduced 
to the above more simple one by dividing the numerator and 
denominator by 1 + £(7^ — Tj). Bessel adopted for r^ the value 
50° F. by Bradley's thermometer; but as this thermometer was 
found to give 1°.25 too much, the normal vahie of the tables is 
r^ = 48°.75 F. Hence, if r,/, or c denote the temperature indi- 
cated by the external thermometer, according as it is Reaumur, 
Fahr., or Cent, we have* 

180 + 16.75 X 0.36438 
^ ■" 180 + J r X 0.36438 

— 180 + 16 .75 X 0.36438 

"" 180 + (/ — 32)" X 0.36438 / (^29) 

_ J180 + 16.75 X 0.36438 
"" ~i80"4- |<^ X 0^6438 

The tables constructed according to these formulie give the 
values of log 5, log T^ and log y^ with the arguments barometer, 
attached thermometer, and external thermometer respectively, 
and the eonii)utation of the true refraction is rendered extremely 
simple. An example has already been given in Art. 107. 

118. In the preceding discussion we have omitted any con- 
sideration of the hygrometric state of the atnios]>here. The 

* TabuUe Regiomontantty p. LXII. 


refractive power of aqueous vapor is greater than that of at- 
mospheric air of the same density, but under the same pressure 
its density is less than that of air ; and Laplace has shown that 
" the increase of the greater refractive power of vapor is in a 
great degree compensated by its decrease of density/** 

119. Refraction (able iciih the argument true zenith distance. — ^When 
the true zenith distance ^ is given, we may still find the refrac- 
tion from the usual tables, or Col. A of Table 11., where the 
apparent zenith distance z is the argument, by successive ap- 
proximations. For, entering the table with ^ instead of Zy we 
shall obtain an approximate value of ?•, which, subtracted from i^, 
will give an approximate value of z ; with this a more exact 
value of r can be found, and a second value of z, and so on, until 
the computed values of r and z exactly satisfy the equation z = 
^ — r. But it is more convenient to obtain the refraction directly 
with the argument ^. For this puq^ose Col. B of Table 11. gives 
the quantities a', j1', ^', which are entirely analogous to the a. Ay 
and >l, so that the refraction is computed under the form 

r = a'y9^>A' tan C (230) 

where /9 and y have the same values as before. 

The values of a', ^', and ^' are deduced from those of a. Ay 
and >l after the latter have been tabulated. They are to be so 
determined as to satisfy the equations 

a,3^r^ tan z = a'^^'r^' tan C (231) 

J = C — a'^S^V^' tan C (232) 

and this for any values of ^ and x- Let {z) denote the value of z 
which corresponds to ^ when /9 = 1, y = 1\ that is, when the 
refraction is at its mean tabular value. The value of {z) may be 
found by successive approximations from Col. A,, as above ex- 
plained. Let (a), {A)y (/), and (r) denote the correspondhig 
values of a. Ay ly r. We have 

(r) = (a) tan (z) = a tan C 

Iz) = Z — a tan C 

whence, by (232), 

♦ Mie, CH. Book X. 


r = (^) — a! tan C (;5^>^' — 1) 
But, taking Napierian logarithms, we have 

and hence, e being the Napierian base, 

/S^>A' = e ^'^^ + A''r = 1 + (A' l? + k' Ir) + &c. 

where, as /9 and ;* differ but little from unity, tlie higher powers 
of A'l^ + k'ly may be omitted. Hence 

z = {z) — (r)iA'ip + X^lr'] 

Now, taking the logarithm of (231), we have 

I (a tan z) + Alp + Xlr = l (a' tan C) + A'lp + I'ly 

The first member is a function of z^ which we may develop as a 
function of (z) ; for, denoting this first member by/?, and putting 

y = - (r) [A'//3 + ^7r] 

we have z = (z)+ y> ^^^ hence 

fz =f iiz) + y] =/ {z) + ^^^ y + &c., 

where we may also neglect the higher powers of y. But since 
f{z) is what/r becomes when z = (z)y and consequently -4 = (-4), 
/ — {X\ we have 

fir) = I [(O tan (^)] + (4) I? + (^) /r 

d^jTfr)^ _ dl[iia) tan a-)] _ d [(a\ tan^)]_ ^ _1_ J<r) 
d(z) ~~ ci (c) "" (a) tan (c) t/ (-:) " (r) ii(c) 

Hence we have 

fz = . I [fa) tan (r)] + (.1) //? + (X) /r - Jj[^^] [4' //5 + x' /r] 

^:/[o'tan:] + A'l? -\-yir 

or, since (a) tan [z) ~ a' tan J, 


Since this is to be satisfied for indeterminate values of ^ and y^ 
the coefficients of l^ and ly in the two members must be equal; 
and therefore 

/ = 

rf(r) ) (233) 

and also 

a' = (a)*'^"(^> 

tan C 

All the quantities in the second members of these formulae may 
be found from Column A of Table 11., and thus Column B may 
be formed.* 
If we put 

we shall now find the refraction under the form 

r = k' tan C 

120. To find the refraction of a star in right ascension and decli- 

The declination d and hour angle t of the star being given, 
together with the latitude (p of the place of observation, we first 
compute the true zenith distance ^ and the parallactic angle q 
by (20). The refraction will be expressed under the form 

r = k' tan C 
in which 

The latitude and azimuth being here constant (since refrac- 
tion acts only in the vertical circle), we have from (50), by put- 

♦ See also Bessel, Astronomisehe Unterauchunyen^ Vol. I. p. 159. 



ting rf^ = 0, rf^ = 0, df = r = A' tan r^di = — dd, (a = star's 
right ascension), 

d9 = 

cos dda = 

— k' tan C cos q 

— /:' tan C sin q 



which are readily computed, since the logarithms of tan ^ cos q 
and tan ^ sin q will already have been found in computing ^ by 
(20). The value of log /j' will be found from Table 11. Column 
B, with the argument ^. 

The values of do and da thus found are those which are to be 
algebraically added to the apparent declination and right ascen- 
sion to free them from the eftect of refraction. 

The mean value of A' is about 57", which may be employed 
when a very precise result is not required. 

Fig. 17. 


121. The dip of the horizon is the angle of depression of the 
visible sea horizon below the true horizon, arising from the ele- 
vation of the eye of the observer above the level of the sea. 
Let CZy Fig. 17, be the vertical line of an obser\'er at -4, 

whose height above the level of the 
sea is AB, The plane of the true ho- 
rizon of the observer at A is a plane 
at right angles to the vertical line 
(Art. 3). Let a vertical plane bo 
passed through CZ, and let BTD bo 
the intersection of this plane with the 
earth's surface regarded as a sphere. 
All its intersection with the horizon- 
tal plane. Draw ATIP in this plane, 
tan«cent to the circular section of the 
earth at T, Disregarding for the pre- 
sent the ettect of the atmosi>here, T'will 
be the most distant point of the surfai'C visible fn>m A, If we 
now conceive the vertical plane to revolve about ("^^as an axis, 
AH will generate the ]»lane of the celestial horizon, while AW 
will generate the surface of a cone touching the earth in the 
small circle called the visible horizon; and the angle HAH' 
will be the dip of the horizon. 


122. To find the dip of the horizortj neglecting the atmospheric refrac- 
tion. Let 

X = the height of the eye = AB, 
a = the radius of the earth, 
D = the dip of the horizon. 

We have in the triangle CAT, ACT= HAH' = D, and hence 

tanD = -— -- 

By geometry, we have 

AT=VAB X AD = Vx{2a-{'X) 


a ^ a ' \ a / 

As a: is always very small compared with a, the square of the 
fraction — is altogether inappreciable: so that we may take 


tan i) = J^ (235) 

123. To find the dip of the horizon, having regard to the atmospheric 

The curved path of a ray of light from the point T, Fig. 18, 
to the eye at A, is the same as that 
of a ray from A to T; and this is ^l^' ^^* 

a portion of the whole path of a 
ray (as from a star S) which passes 
through the point A, and is tangent 
to the earth's surface at T The ^' 
direction in which the observer at h' 
A sees the point T is that of the 
tangent to the curved path at A, or 
AH'; the true dip is therefore the 
angle HAH', and is less than that found in the preceding article. 
It is also evident that the most distant visible point of the earth's 

174 DIP OP THE horizon; 

surface is more remote from the observer than it would be if 
the earth had no atmosphere. 

Now, recurring to the investigation of the refraction in Art 
108, we observe that the angle HAH' is the complement of 
the angle of incidence of the ray at the point A^ there denoted 
by i; and it was there shown that if y, [Xy and i are respectively 
the normal, the index of refraction, and the angle of incidence 
for a point elevated above the earth's surface, while a, /i^ and z 
are the same quantities at the surface, we have 

• • 

9 /£ sm t = a A<o sm ^ 
But in the present case we have z = 90° ; and hence, putting 

D' = the true dip = 90^ — i 
q =a -{- X 

we have 

sin t = cos iJ = — . = — 1 1 H I 

M a -\- X m\ a I 


Developing and neglecting the square of — as before, 

cosi)' = ^(l-|^) (236) 

which would suffice to determine D' when /^ and fi have been 
obtained from the observed densities of the air at the observer 
and at the level of the sea. But, as JD' is small, it is more con- 
venient to determine it from its sine; and we may also intro- 
duce the density of the air directly into the formula by putting 
(Art. 110), 

M \ 1 + 4^^ 
Substituting the value of a from (178), namely, 

_ 2kdo 
* ""1 + 4^^0 

we may give this the form 



8 1-i 

which, by neglecting the square of the second tenn, gives 

^=> + -('-t) 

Hence, still neglectiug the higher powers of a and — , as well as 
their product, we have 

8mi>' = >/l— co8*i>' = ^|^— 2a/l— ^U (237) 

which agrees with the formula given by Laplace, JUffc. CSl. 
Hook X. 

For an altitude of a few feet, the difference of pressure will 
not sensibly affect the value of i)', and may be disregarded, 
especially since a very precise determination of the dip is not 
possible unless we know the density of the air at the visible hori" 
Z071, which cannot usually be observed. We may, however, 
assume the temperature of the water to be that of the lowest 
stratum of the air, and, denoting this by r^, while r denotes the 
temperature of the air at the height of the eye, we have [mak- 
ings = Pq in (171)], approximately, 

in which for Fahrenheit's thermometer e = 0.002024. Hence 

I sin' D ) 

where D is the dip, computed by (235), when the refraction is 
neglected, the sine of so small an angle being put for its tan- 
gent. If we substitute the values a = 0.00027895, sin D = 
D sin 1", and $ = 0.002024, this formula becomes 


_pr_j 24021 (r~r,) 

in which D is in seconds. If JD is expressed in minutes in the 
last term, it will be sufficiently accurate to take 

2)' = 2)-400x^^ (238) 

This will give D' = D when r = r,,, as it should do, since in 
that case the atmosphere is supposed to be of uniform density 
from the level of the sea to the height of the observer. If 
^ < ^o» ^'® have D' '^ D. In extreme cases, where r is much 
greater than r^,, we may have JD' < 0, or negative, and the visible 
horizon will appear above the level of the eye, a phcnomenou 
occasionally observed. I know of no observations sufficiently 
precise to determine whether this simple formula, deduced from 
theoretical considerations, accurately represents the observed 
dip in every case. 

124. If, however, we wish to compute the value of D' for a 
mean state of the atmosphere without reference to the actually 
observed temperatures, we may proceed as follows : In the equa- 
tion above found, 

cos2)'= - . 

we may substitute the value 

//I \" + ^_ a 
\/iJ ~a+x 

which is our first hypothesis as to the law of decrease of density 
of the strata of the atmosphere, Art. 109. This hypothesis will 
serv'c our present i)urpo8e, provided 7i is so determined as to 
rei)re8cnt the actually observed mean horizontal refraction. We 
have, then. 


and developing, neglecting the higher powers of—, 


n X 

COS D' = 1 — 

n + 1 a 

Bin D' = \l — r-T • — = tan D \l — ^T 

\n4-l a \n + 1 

+ 1 a \n + 




To determine n, we have by (160), reducing Vq to seconds, 

n = 

(ro sin 1")* 

where, for Barom. 0*.76, Therm. 10°C., which nearly represent 
the mean state of the atmosphere at the surface of the earth, we 
have 4k d^ = 0.00056795, and r^ = 84' 30" (which is about the 
mean of the determinations of the horizontal refraction by dif- 
ferent astronomers) ; and hence we find 

n = 5.639, J-JL. = 0.9216 = 1 — 0.0784 

\n + 1 

D' = D — .0784i) (239) 

The coefficient .0784 agrees very nearly with Delambre's value 
.07876, which was derived from a large number of observations 
upon the terrestrial refraction at different seasons of the year. 
To compute D' directly, we have 

sm 1" ^ d 

If a: is in feet, we must take a in feet. Taking the mean value 

a = 20888625 feet, and reducing the constant coefficient of >/F, 
we have 

D' = 58".82 i/x in feet. (240) 

Table XI., Vol. IL, is computed by this formula. 

Vol. L— 12 


125. To find the distance of the sea horizon^ and the distance of an 
object of known height jxist visible in the horizon. — The small portion 

TAy Fig. 19, of the curved path of a ray of 
^**' ^^' ligl^t, may be regarded as the arc of a circle; 

and then the refraction elevates A as seen 
from T as much as it elevates T as seen 
from A. Drawing the tangent TP, the ob- 
server at T would see the point A at P; 
A'^Bf \ \ ai^j if the chord TA were drawn, the angle 

PTA would be the refraction of A. This 
refraction, being the same as that of T as 
seen from A^ is, by (239), equal to .07842). In the triangle 
TPAy TAP is so nearly a right angle (mth the small elevations 
of the eye here considered) that if we put 

x^ = AP 

we may take as a sufficient approximation 

x^ = TAy^ tan PTA = a tan D X .0784 tanD 

But we have a tan^JD = 2a:, and hence 

x^ = .1568 a; 

d = the distance of the sea horizon, 
we have 

PT=y'i2CB + PB) X PB 
or, nearly, 

d = \/2a (.r + .rj = v/2.3136aa: 

If X is given in feet, we shall find d in statute miles by dividing 
this value by 5280. Taking a as in the preceding article, we 

and, therefore. 

V2Mma _ j_3j^ 

d (in statute miles) t= 1.317 |/x in I'ect. (241) 

If an ol)server at -4' at the height A'B' — x' sees the object 
-4, whose height is a:, in the horizon, he must be in the curve do- 


scribed by the ray from A which touches the earth's surface at 

T. The distance of A* from Twill be = 1.317 i/x^', and hence 

the whole distance from A to A^ will be = 1.317 {Vx + Vx^). 

Tlie above is a rather rough approximation, but yet quite as 
accurate as the nature of tlie problem requires ; for the anoma- 
lous variations of the horizontal refraction produce greater 
errors than those resulting from the formula. By means of this 
formula the navigator approaching the land may take advantage 
of the first appearance of a mountain of known height, to deter- 
mine the position of the ship. For this purpose the formula 
(241) is tabulated with the argument "height of the object or 
eye ;" and the sum of the two distances given in the table, cor- 
responding to the height of the object and of the eye respect- 
ively, is the required distance of the object from the observer. 

126. To find the dip of the sea at a given distance from the observer. 
— ^By the dip of the sea is here understood the apparent depres- 
sion of any point of the surface of the water nearer than the 
visible horizon. Let T, Fig. 20, be such a 
point, and A the position of the observer. 
Let TA' be a ray of light from T, tangent 
to the earth's surface at T, meeting the ver- 
tical line of the observer in A'. Put 

D"= the dip of T sls seen from A, 

d = the distance of T in statute miles, 

X = the height of the observer's eye in feet = AB^ 

x' = A'B. 

We have, by (241), 


and the dip of Ty as seen from A'^ is, therefore, by (240), 

= 58".82 i/F = 44".66 d. 

Now, supposing the chords 7!A, TA' to be drawn, the dip of T 
at A exceeds that at A' by the angle A TA'^ very nearly ; and 
we liave nearly 

aiii'lo ATA = X = 

TA' sinl" 5280 <« sin 1" 



5280 d Bin 1" 
Substituting the value of x' in terms of rf, 

D" = 22'M4 d + 39".07 - (x being in feet and d in statute 

miles). (242) 


If rf is given in sea miles, we find, by exchanging d for 7~rf, 

D" = 25".65 d + 33".73 -(x being in feet and d in sea 


miles). (243) 


The value of D" is given in nautical works in a small table 
with the arguments x and d. The formula (243) is xQty nearly 
the same as that adopted by Bowditch in the Pi^actical Navigaicr. 

127. At sea the altitude of a star is obtained by measuring its 
angular distance above the visible horizon, which generally 
appears as a well-defined line. The observed altitude then 
exceeds the apparent altitude by the dip, remembering that by 
apparent altitude we mean the altitude referred to the true 
horizon, or the conipleinent of the apparent zenith distance. 
Thus, A' being the observed altitude, h the apparent altitude, 

or, when the star has been refeiTcd to a point nearer than the 
visible horizon, 


128. In order to obtain by observation the position of the 
centre of a celestial body which has a well-defined disc, we 
observe the position of some point of the limb and deduce that 
of the centre bv a suitable application of the angular scmi- 
diametcr of the bodv. 

I shall here consider onlv the case of a si)herical bodv. The 

ft I • 

api^arent outline of a jilanct, whether s]»herical or spheroidal, 
and whether fully or partially illuminated by the sun, will be 



Fig. 21. 

discussed in connection \vith the theor}' of occultations in 
Chapter X. 

The angular semidiametcr of a spherical body is the angle 
subtended at the place of obsei-v^ation by the radius of the disc. 
I shall here call it simply the seniidiameter, and distinguish the 
linear seniidiameter as the radius. 

Let O, Fig. 21, be the centre of 
the earth, A the position of an ob- 
server on its surface, M the centre 
of the observed body; 0J5, AB'^ 
tangents to its surface, drawn from 
and ^. The triangle OEM re- 
volved about OM as an axis will de- 
scribe a cone touching the spherical 
body in the small circle described 
by the point J5, and this circle is the 
disc whose angular seniidiameter at 
is MOB. Put 

S = the geocentric semidiametcr, MOB, 
S' = the apparent semidiametor, MAB\ 
J,J' = the distances of the centre of the body from the centre of 
the earth and the place of observation respectively, 
a = the equatorial radius of the earth, 
a' = the radius of the body, 

then the right triangles OMBy AMB' give 

. „ a! 
sm o = — 


sin S' = 



But if 

T = the equatorial horizontal parallax of the body, 

we have, Art. 89, 

and hence 


SHI TT = — 


sm aS = — sm t: 

sin S' = — sin S 


or, with sufficient precision in most cases, 


S'= — 8 



The geocentric semidiameter and the horizontal parallax have 


therefore a constant ratio = — . For the moon, we have 



- = 0.272956 (247) 


a8 derived from the Greenwicli observations and adopted by 
Hansen {Tables de la Lwie, p. 39). 

If the body is in the horizon of the observer, its distance from 
liim is nearly tlic same as from the centre of the earth, and hence 
tlie geocentric is frequently called the horizontal semidiameter; 
but this designation is not exact, as the latter is somewhat greater 
than the former. In the case of the moon the difterence is 
betAveen 0".l and 0".2. See Table XII. 

If the body is in the zenith, its distance from the observer la 
less than its geocentric distance by a radius of the earth, and the 
apparent semidiameter has then its greatest value. 

The apparent semidiameter at a given place on the earth's 
surface is computed by the second equation of (245) or (246), in 

which the value of — is that found by (104) ; so that, putting 2 — 

the true (geocentric) zenith distance of the body, ^' = the appa- 
rent zenith distance (affected by parallax), A — its azimuth, 
f — <p' the reduction of the latitude, we have, (by (111) and (104), 

Y =. {<p — if') Qo?^ A \ 

8in(C —r) ^ 

129. This last formula is rigorous, but an approximate fonnnla 
for computing the difference aV — aS will sometimes be convenient. 
In (103) we may put 

cos (i p — if') _ 

cos /'COS ] (C' — C) 

without sensible error in computing the very small diflerence in 
question ; we thus obtain 


^, 1 _ ^ sin - cos [} (C 4- :) — f\ 



m = p&mi: COS [i (C' + C) — r] (249) 

we have 

J 1 

J' 1 — m 

= 1 + m + m* + &o. 

and hence, smce the third power of m is evidently insensible, 

S' — S = Sm + &n» (250) 

which is practically as exact as (248). The value of ^' required 
in (249) will be found with sufficient accuracy by (114), or 

C' — C = /o7r8in(C' — r) 

The quantity S' — S is usually called the augmentation of the 
semidiameter. It is appreciable only in the case of the moon. 

130. If we neglect the compression of the earth, which will 
not involve an error of more than 0".05 even for the moon,* we 
may develop (250) as follows. Putting /> = 1 and 7* = in (249), 
we may take 

m = sin ff cos i (C + C) 

= sin n cos [C — } (c' — 0] 

= sin TT cos C' + } sin tt sin (C' — C) sin C 

= sin n cos C' + i sin'r sin' C' 

which substituted in (250) gives, by neglecting powers of sin n 
above the second, 

/S' — S = iS sin TT COS C + i S sin^w sin' C + S sin'Tr cos* C 
= 8 sin TT cos C' + }t S sin'Tr -\- k S sin*7r cos* C' 

But we have 

„ a' a* sin it 

« a sin 1" 

* The greatest declination of the moon being less than 30°, it can reach great 
altitudes only in low latitudes, where the compression is less sensible. A rigorous 
ioTestigation of the error produced by neglecting the compression shows that the 
maximum error is less than 0".06. 


and if we put 

A = iL sin 1", log h = 5.2495 

we have sin t: ^ hS, which substituted above gives the follow- 
ing formula for computing the augmentation of the moon's 

S'—S = hS'cos:'+ih^S*+ JA«/S»cos«C' (251) 

Example.— Find the augmentation for C' = 40°, S = 16' 0" 
= 960". 

Ist term = 12^.54 
2d « = .14 
3d " = 0. 08 
iS' — S = 12 .76 
log 3d term 8.914 

The value of S' — S may be taken directly from Table XTE. with 
the argument apparent altitude = 90® — ^'. 

131. If the geocentric hour angle (t) and declination {d) are 
given, we have, by substituting (137) in (245), 

sin S' = sin S ^1" ^// ~ ^} (252) 

sm (^ — y) ^ ' 

for which y and 5' are to be determined by (134) and (136), or 
with sufficient accuracy for the present purpose by the formulae 

tan <p' 

' cos t 

log /S« 5.9645 

log S^ 8.947 

log h 5.2495 

log J A« 0.198 

log cos C 9.8843 

log 2d term 9.145 

log Ist term 1.0983 

log cos' C 9.769 

^ — ^ = 

p n sin ^' sin (d — y) 
sin ^ 

132. To find the contraction of the vertical semidiameter of the sun 
or moon produced by atmospheric refraction. 

Since the refraction incroases with the zenith distance, the 
refraction for the centre of the sun or the moon will be greater 
than that for the upper limb, and that for the lower limb will be 
greater than that for the centre. The apparent distance of the 


limbs is therefore diminished, and the whole disc, instead of 
being circular, presents an oval figure, the vertical diameter of 
which is the least, and the horizontal diameter the greatest. 
The refraction increasing more and more rapidly as the zenith 
distance increases, the lower half of the disc is somewhat moro 
contracted than the upper half. 

The contraction of the vertical semidiameter may be found 
directly from the refraction table, by taking the difterence of 
the refractions for the centre and the limb. 

Example. — The true semidiameter of the moon being 16' 0", 
and the apparent zenith distance of the centre 84°, find the con- 
traction of the upper and lower semidiameters in a mean state 
of the atmosphere (Barom. 30 inches, Therm. 50° F.). We find 
from Table I. 

For apparent zcn. dist. of centre, 84° 0' Eefr. = 8' 28".0 

" approx. « upper limb, 83 44 « = 8 9 .4 

" " " lower " 84 16 " =8 48 .1 


Approx. contraction upper semid. = 8' 28".0 — 8' 9".4 = 18".6 
" " lower " = 8 48 .1 — 8 28 .0 = 20 .1 

These results are but approximate, since we have supposed the 
apparent zenith distance of the limb to difter from that of the 
centre by the true semidiameter, whereas they difter only by the 
apparent or contracted semidiameter. Ilence we must repeat as 
follows : 

App. zen. dist. upper limb = 83° 44' 18".6 Eefr. = 8' 9".7 

" " lower " = 84 16 39 .9 " = 8 47 .7 

Contraction of upper semid. = 8' 28".0 — 8' 9".7 = 18".3 
" lower " = 8 47 .7 — 8 28 .0 = 19 .7 

Obser\'ations at great zenith distances, where this contraction 
is most sensible, do not usually admit of great precision, on 
account of the imperfect definition of the limbs and the uncer- 
tainty of the refraction itself. It is, therefore, sufliciently exact 
to assume the contraction of either the upper or lower semi- 
diameter to be equal to the mean of the two. In the above 
example, which oftei's an extreme case, if we take the mean 



19" as the contraction for either seniidiameter, the error vnll 
be only 0".7, which is quite within the limit of error of observa- 
tions at such zenith distances. 

133. To find the contraction of any inclined seniidiameler^ produced 

by refraction. 
Let My Fig. 22, be the apparent place of the sun's or tlie 

moon*s centre; ACBD^ a circle described 
with a radius 3IA equal to the true semi- 
diameter, will represent the disc as it would 
appear if the refraction were the same at 
all points of the limb. The jKjint j4, how- 
J> ever, being less refracted than 3/, will ap- 
pear at A\ P at P', &c. ; while iJ, being 
more refracted than J/, appears at B\ The 
contraction is sensible only at great zenith 
distances, where we may assume that AM 

and PP'Ey small portions of vertical circles drawn through A 

and P, are sensibly parallel. If then we put 

S = the true vertical somidiametcr = AMj 

S^--= the contracted vert, scmid. = -4' J/, 

aS, = the contracted inclined semid. = J/P', which makes an 
angle q with the vertical circle, 
AiS^j = the contraction of the vortical semid. = S — S^ 
aJS^--zl^ the contraction of the inclined semid. = aS' — JS^ 

we shall have 

S^ cos q -- P^E = the difference of the apparent zenith distances 
of M and P', 
S^ Z.Z the difference of the app. zen. dist. of J/ and A'. 

Xow, the difference of the refractions at M and A^ is AA\ and 
the diflcrence of the refractions at J/ and P' is PP' ; and, since 
these small difierences are nearly proportional to the difierences 
of zenith distance, we have 

S, : *% cos q --: AA' : PP' 


s -'-^^^? 



The small triangle PFP^ may be regarded as rectilinear and 
right-angled at F; whence 

FP' = PP' X cos q 

If we put /Si for S^ in the second member, the resulting value of 
dSq will never be in error 0".2 for zenith distances less than 85^, 
and it sufKces to take 

AS^ = JS, C08« q (253) 

This formula is sufficiently exact for all purposes to which we 
shall have occasion to apply it. 

134. To find the contraction of the horizontal seniidiameter. — The 
formula (253) for q = 90° makes the contraction of the hori- 
zontal semidiameter = 0. This results from our having assumed 
that the portions of vertical circles drawn through the several 
points of the limb are parallel, and this assumption de- 
parts most from the truth in the case of the two ver- 
tical circles drawn through the extremities of the 
horizontal diameter. To investigate the error in tliis 
case, let ZM^ Fig. 23, be the vertical circle drawn 
through the centre of the body, ZM' that drawn 
through the extremity of the horizontal semidiameter 
MM'. In consequence of the refraction, the points M 
and M' appear at iVand iV'. If we denote the zenith -jf 
distances of M and N by ^ and z, those of M' and N' 
by C' a^id 2:', the refraction MN may be expressed as a func- 
tion either of z or of i^. Art. 107, and we shall have 

r = k tan z = k' tan C 

where k and A-' are given by the refraction table with the argu- 
ments z and ^. The zenith distance of the point M' difters so 
little from that of M that the values of k and k' will be sensibly 
the same for both points, and we shall have for the refraction 


r' = k tan 2' = A' tan C 


These two equations give 

tan z tan C 

tan if tan C' 
But if the triangle ZNN' is right-angled at Ny we have 

tan z 

cos Z = 

tan ^ 

and hence, also, 

ry tan C 
cos Z =i 


Therefore the triangle ZMM' is also right-angledj and it gives 

^ _ tan 5 tan^' 

tan Z = -T—. — -^ = — . — 
sm {z + r) sin z 

in which S = MM' and 5' = NW. Hence 

tan 5f sin (2: + r) , . 

7 — c-/ = — -. — ■ — - = cos r + sin r cot z 

tan 6' sin 2: ' 

or, very nearly, 

%^ = 1 + r sin 1" cot 2 = 1 + ^ sin 1" 

Hence the contraction of the horizontal semidiameter is ex- 
pressed by the following formula : 

S— S' = S' k sin 1" 

In the zenith, the mean value of log k is 1.76156; at the zenith 
distance 85^ it is 1.71020. For S' =-- 10', therefore, the contrac- 
tion found by this formula is 0".27 in the zenith, and 0".24 for 
85°. Thus, /or all zenith distahccs less than So^ the contraction of 
the horizontal semidiameter is very nearly constant and equal to one- 
fourth of a second, 

AVhen the body is in the horizon, we have k = root * = 0, 
and hence *S' — *S'' - 0, which follows also from the sensible 
parallelism of the vertical circles at the horizon. 




135. It is important to observe a proper order in the applica- 
tion of the several corrections which have been treated of in this 

The zenith distance of any point of the heavens observed wHth 
any instrument is generally affected with the index error and 
other instnimental errors. These errors will be treated of in 
the second volume ; here we assume that they have been duly 
allow^ed for, and we shall call "observed" zenith distance that 
which would be obtained with a perfect instrument, and shall 
denote it by z. 

In all cases the first step in the reduction is to find the refrac- 
tion r (=ay9^;'^ tan z) with the argument z^ and then z + r is the 
zenith distance freed from refraction. 

1st. In the case oi a fixed star^ 

C = -2r + r 

is at once the required geocentric zen. dist. 

2d. In the case of the moon., the zenith distance observed is 
that of the upper or lower limb. If S is the geocentric and S' 
the augmented semidiameter found by Art. 128, 129, or 130, 

is the apparent zenith distance of the moon*s centre freed from 
refraction, and affected only by parallax, and, consequently, it is 
that which has been denoted by the same symbol in the discus- 
sion of the parallax. With this, therefore, we compute the 
parallax in zenith distance, ^' — (^, by Art. 95, and then 

is the required geocentric zenith distance of the moon's centre. 

To compute S' by (248), (250), or (251), we must first know C'; 
but it will suffice to employ in these formulee the approximate 
value ^' = 2 + r dz /S. 

AVe can, however, avoid the computation of aS', when extreme 
precision is not required, by computing the parallax for the 
zenith distance of the limb. Thus, putting ^' = 2 + r, and 


compnting C' — C ^^Y ^^- ^'% ^^^ quantity C — C "" (C' "~ C) ^* 
the geocentric zenith distance of the limb; and therefore, ap- 
plying the geocentric seniidiameter, ^ ±: S is the required geo- 
centric zenith distance of tlie moon's centre. This process 
involves the error of assuming the horizontal parallax for the 
limb to be the same as that for the moon's centre. It can easily 
be shown, however, that the error in the result will never amount 
to 0".2, which in most cases in practice is unimportant. The 
exact amount will be investigated in the next article. 

3d. In the case of the s^m or a planet^ when the limb has been 
obser\'ed, the process of reduction is, theoretically, the same as 
for the moon ; but the parallax is so small that the augmentation 
of the semidiameter is insensible. We therefore take 

V = z + r ± S 

and then, computing the parallax by Art. 96, or even by Art. 90, 
^ — ^' — (C' ~ C) 5^ ^^^ ^^"^ geocentric zenith distance. 

If a point has been referred to the sea horizon and the 
measured altitude is H^ then, D behig the dip of the horizon, 
A' = II— D is properly the observ^ed altitude, and z = 90® — A' 
the observed zenith distance, with which we proceed as above. 

136. The process above given for reducing the observed zenith 
distance of the moon's limb to the geocentric zenith distance of 
the moon's centre, is that which is usually employed; but the 
whole reduction, exclusive of refraction, may be directly and 
rigorously computed as follows. Putting 

C' = c + r = the apparent zenith distance of the moon's limb 

corrected for refraction, 
C = the geocentric zenith distance of the moon's centre, 

then, *S' being the augmented semidiameter, we must substitute 
^' ±. *S" for ^' in the formula* for parallax, and, by (101), we 

/ sin (C' ± aS') =:= sin C — f» sin r cos (9- — 9?') tan y 
f cos (C' d: S*) — - cos C — /> sin r cos (^p — /) 

Multiplying the first of these by cos ^', the second by sin ^', and 
subtracting, we have 


±:f Bin JS' = — sin (C — C) + ^- ^^^^ ^--^ sm (C' — - /') 

cos y 

in which /= --. By (245) we have also 

/ sin /S' = sin S 
and hence the rigorous fonnula 

sin (C - C) = /> sin TT sin (C - r) ^^^ (^ ~ ^0 zp gin ^r 

cos y 

for which, however, we may employ with equal accuracy in 

sin (C' — = /> sin r sin (C — r) =F sin S (254) 

iu which, J. being the moon's azimuth, we have 

^ = (^ — ^') cos A 

If we put (Art 128) 

A = ~ = 0.272956 

we have sin S=k sin ;:, and (254) may be written as follows: 

sin (C — C) = |> sin (C' -- r) =F A] sin ;: (255) 

For convenience in computation, however, it will be better to 
make the following transformation. Put 

Hinp = p sin 7t sin (C' — ;') (256) 

then (254) becomes 

sin (C' — C) = sin^ qi sin S 

= Bin (p ::f S) -\- sin p (1 — cos S) =p sin S (1 — cOBp) 
= sin ( j? qi /S) + 2 sin p sin* J ^' qi 2 sin >S^ sin* ip 

where the last t\vo terms never amount to 0''.2, and therefore the 
formula may be considered exact under the form 

sin (C — C) = sin (p zp S) qz i (p zp S) sin 1" sin p Bin S 
Since C' — C and pzpS differ by so small a quantity, there will 


be no appreciable error in regarding them as proportional to 
their sines ; and hence we have 

C — : =p T /S H= i (/> =H 'S) 8in;> sin 5 (257) 

the upper signs being nsed for the upper limb and the lower 
signs for the lower limb. 

In this formula, p is the parallax computed for the zenith 
distance of the limb, and the small term i(/> =P S)sin p sin S may 
be regarded as the connection for the error of assuming the 
parallax of the limb to be the same as that of the centre. 

Example. — ^In latitude (p = 38° 59' N., given the observed zenith 
distance of the moon's lower limb, z = 47° 29' 58", the azimuth 
A = 33° 0', Barom. 30.25 niches. At. Therm. 65° F., Ext Therm. 
64° F., Eq. hor. par. ;: = 59' 10".20 ; find the geocentric zenith 
distance of the moon's centre : 

(Table III.) (^ — ^') = 11' 15" » = ^7^ 29'68''.00 

log (9 _ ^') = 2.8293 (Table II.) r = 1 2 .27 

log cos A 9.9236 C' = 47 31 .27 

log }' 2.7529 y = ^ 26 . 

(Table III.) log p 9.999428 T — y = ^7 21 34 . 

log sin T 8.235806 

log sin (;' — >') 9.800652 

log sin /> 8.101886 p= 43' 28". 09 

log sin T 8.235806 S = 16 9 .00 

(Art. 128) log (0.272956) 9.436003 p ^ S= 59 87 .09 

log sin S 7.r,71899 i (p + S)8in p sin S = Al 

log sin p sin S 5.7739 C — C = 59 37 .20 

log {p 4- S) 3.5535 

log J 9.6990 C = 46® Sr 23''.07 

log Kp-tS) sinp sin S 9.0264 

It IS hardly nceessarj- to observe that if the geocentric zenith 
distance of the centre of the moon or other body is given, the 
api»areiit zenith distance of the limb affected by parallax and 
refraction will be deduced by reversing the order of the steps 
above exjilaincd. 

If altitudes are given, we may employ altitudes throughout 
tlie ooniiaitation, i)utting everjwhere 90° — ^, &c. for ^, &c., and 
making the necessary obvious modilications in the formulae. 




137. We have seen, Art. 55, that the local time at any place 
is readily found when the hour angle of any known heavenly 
body is given. This hour angle is obtained by observation, but, 
a direct measure of it being in general impracticable, we must 
have recourse to observations from which it can be deduced. 

The observer is supposed to be provided with a clock, chro- 
nometer, or watch, which is required to show the time, mean or 
sidereal, either at his own or at some assumed meridian, such as 
that of Greenwich. 

The clock correction* is the quantity which must be added alge- 
braically to the time shown by the clock to obtain the correct 
time at the meridian for which the clock is regulated. K we put 

T = the clock time, 
T' = the true time, 
A T = the clock correction, 
we have 

T' = T + aT 
or aT = T—T (258) 

and the clock correction will be positive or negative^ according as 
the clock is slow or fast. It is generally the immediate object of 
an observation for time to determine this correction. At the 
instant of the observation, the time T is noted by the clock, 
and if this time agrees vnth the time T' computed from the 
observation, the clock is correct ; otherwise the clock is in error, 
and its correction is found by the equation ^T= T' — T. 

The clock rate is the daily or hourly increase of the clock cor- 
rection. Thus, if 

* For breTitjr, I shall use dock to denote anjr time-keeper. 
Vol. L— 13 


aT^ = the clock correction at a timo T^, 

we have 

dT = the clock rate in a unit of time, 

aT = A 7; + ^r (T — TJ (259) 

where T — Tq must be expressed in days, hours, &c., according 
as 5 y is the rate in one day, one hour, &c. 

When, therefore, the clock correction and rate have been 
found at a certain instant T^, we can deduce the true time from 
the clock indication 7" (or "clock face," as it is often called) 
at any other instant, by the equation 

T' = T + aT^ + dT^T--- 2;) (260) 

If the clock correction has been determined at two different 
times Tq and Tj the rate is inferred by the equation 

dT=^-^^^^ f261) 

But these equations are to be used only so long as we can 
regard the rate as constant. 

Since such uniformity of rate cannot be assumed for any great 
length of time, even with the bcvst clocks (although the perform- 
ance of some of them is really suii)rising), it is proper to make 
the intcn'al between the observations for time so small that the 
rate may be taken as constant for that inter\'al. The length of 
the inten^al will depend upon the character of the clock and the 
degree of accuracy required. 

Example. — At noon. May 5, the correction of a mean time 
clock is — IC" 4?.30 ; at noon. May 12, it is — 16** 13'.50 ; what is 
the mean time on May 25, when the clock face is 11* IS* 12*.6, 
supposing the rate to be unifonn ? 

May 5, corr. = — 16" 47*.30 
" 12, '' = — 16_13^50 

Rate in 7 days = + BsTsO 
5r= + 4.829 

Taking, then, as our starting point T^ — May 12, 0*, we have 

TIME. 195 

for the interval to 7^= May 25, 11* 13"» 12^6, T-T^ = IS** 11* 
13* 12'.6 = 13*^.467. Hence we have 

A To = — 16- 13'.50 
dT(T-^ To) = + 1 5 .03 

£^T=— 15 8.47 
T = 11* 13"> 12'.60 

T = 10 68 4 .13 

But in this example the rate is obtained for one true mean 
day, while the unit of the interval 13''.467 is a mean day as 
shown by the clock. The proper interval with which to com- 
pute the rate in this case is 13'' 10* 58"' 4M3 = 1»'.457 with 
which we find 

aT, = — 16-13'.50 
dT X 13.457 = + 1 4 .98 

- aT=— 16 8.52 

T = 11* 13«> 12'.60 

T'=10 58 4.08 

This repetition will be rendered unnecessary by always giving 
the rate in a tinit of the clock. Thus, suppose that on June 3, 
at 4* 11"* 12*.35 by the clock, we have found the correction 
+ 2* 10^.14 ; and on June 4, at 14* 17- 49'.82, we have found 
the correction + 2* 19*.89 ; the rate in one hour of the clock will be 

^r=±^— = + 0'.2858 

For practical details respecting the care of clocks and other 
time-keepers, the methods of comparing their indications, &c., 
see Vol. n. ; sec also Chapter VII., ''Longitude by Chronometer.** 
I shall here confine myself to the methods of determining their 
correction by astronomical observation. 

Those methods, however, which involve details depending 
upon the peculiar nature of the instrument with which the ob- 
servation is made, will be treated very briefly in this chapter, 
and their full discussion will be reserved for Vol. 11. 

196 TIME. 


138. At the instant of a star's passage over the meridian, note 
the time Thy the clock. The star's hour angle at that instant 
is = 0*, whence the local sidereal time T' is (Art. 55) 

T' = a = the star's right ascension. 

If the clock is regulated to the local sidereal time, we have, 

aT=o— T 

But if the clock is regulated to the local mean time, we first con- 
vert the Bidereal time a into the corresponding mean time T' 
(Art. 52), and then we have 

aT= T — T 

This, then, is in theory the simplest and most direct method 
possible. It is also practically the most precise when properly 
carried out with the transit instrument. But, as the transit iii- 
strumcnt is seldom, if ever, precisely adjusted in the meridian, 
the clock time 2' of the true meridian transit of a star is itself 
deduced from the observed time of the transit over the instru- 
ment by api)lying proi)er corrections, the theory of which will 
be fullv discussed in Vol. U. 

It will there be seen, also, that the time may be found from 
transits over anv vertical circle. 


130. (A.) Equal nlfititdcs nf a fixed star. — The time of the meri- 
dian transit of a fixed star is the mean between the two times 
when it is at the same altitude east and west of the meridian: so 
tbat the observation of these two times is a convenient substi- 
tute for tbat of the meridian ])assage when a transit instrument 
is not available. The observation is most frequently made with 
the sextant and artificial horizon : but any instrument adapted to 
the measurement of altitudes may be emjiloyed. It is, however, 
not required tbat the instrument should indicate the true alti- 
tude ; it is suflicient if the altitude is tlw i^amc at both ob8er\'a- 


tions. If we use the same instrument, and take care not to 
change any of its adjustments betw-ecn the two observations, we 
may generally assume that the same readings of its graduated 
are represent the same altitude. Small inequalities, however, 
may still exist, which will be considered hereafter.* 

The clock correction will be found directly by subtracting 
the mean of the two clock times of observation from the com- 
puted time of the star's transit. 

Example 1. — March 19, 1856; an altitude of Arcturus east 
of the meridian was noted at 11* 4"^ 5r.5 by a sidereal clock, 
and the same altitude west of the meridian at 17* 21"* SO'.O; find 
the clock correction. 

East 11* 4-51*.5 

West 17 21 30.0 

Merid. transit by clock = T = 14 13 10 .75 
March 19, Arcturus E. A. = o = 14 9 7 .11 

Clock correction = ^T =z — 4 3. 64 

This is the clock correction at the sidereal time 14* 9"* 7*.ll or 
at the clock time 14* 13" 10'.75. 

Example 2. — March 15, 1856, at the Cape of Good Hope, 
Latitude 33° 56' S., Longitude 1* 13'" 56* E.; equal altitudes of 
Spica are observed with the sextant as below, the times being 
noted by a chronometer regulated to mean Greenwich time. 
The artificial horizon being employed, the altitudes recorded are 
double altitudes. 


2 Alt. Spica. 


10*20- 0*.5 

104° 0' 

2* 40" 38*. 

" 20 28. 

« 10 

« 40 10.5 

" 20 55. 

" 20 

" 39 42. 

10 20 27.83 

2 40 10.17 

10 20 27.83 

Merid. Transit, by Chronom. = T = 12 30 19 .00 

The chronometer being regulated to Greenwich time, we 
must compute the Greenwich mean time of the star's transit at 
the Cape (Art. 52). We have 

* For the method of observing equal altitudes with the sextant, see Vol. II., 
























198 TIME. 

Local sidereal time of transit = a = 18* 17"* 87*. 92 
Longitude == — 1 18 56 . 

Greenwich sidereal time = 

March 15, sid. time of mean noon = 

Sid. interval from mean noon = 

Reduction to mean time = 

Mean Or. time of star's -> _ 

local transit / 

Chronometer time of do. = 7* = 

Chronometer correction = A 7* = — 1 46 .42 

140. (B). Equal altitudes of the sun before and after noon. — K the 
declination of the sun were the same at both observations, the 
hour angles reckoned from the meridian east and west would be 
equal when the altitudes were equal, and the mean of the t^'o 
clock times of observation would be the time by the clock at 
the instant of apparent noon, and we should find the clock cor- 
rection as in the case of a fixed star. To find the correction 
for the change of declination, let 

9 = the latitude of tho place of observation, 
d = the sun's declination at apparent (local) noon, 
A(J = tho increase of declination from the meridian to the west 
observation, or the decrease to tho east observation, 
h = tho sun's true altitude at each observation, 
T„ = the mean of the clock times A.M. and P.M., 
aTo = tho correction of this mean to reduce to tho clock time 
of apparent noon, 
t = half tho elapsed time between the observations. 

Then we have 

t -\- ^Tq=^ tho hour angle at tho A.M. observation reckoned 

towards the east, 
t — A 2^^= tho hour angle at the P.M. observation, 
^ — Ao = tho declination at the A.M. " 
a + ArJ = '* " P.M '' 

and, by the first equation of (14) applied to each observation, 

sin li = sin <p sin («J — ao) + cos <p cos (d — a*)) cos {t -\- a T^ 
Bin A = sin ^ sin ((5 + a5) -f cos tp cos {d -f ^^) ^^^ (J — ^ ^») 


If we substitute 

sin (^ ± A^) = sin d cos Ad ±: cos d sin ^d 
cos (d -±2 Ad) = COS 8 COS Ad =p sin d sin Ad 
cos (< ± A To) = COS t COS A To T sin < sin a T^ 

and then subtract the first equation from the second, we shall 


= 2 sin 9 cos d sin Ad — 2 cos f sin d sin Ad cos t cos a T^ 

-|- 2 cos f cos d sin t cos Ad sin a T^ 
whence, by transposing and dividing by the coefficient of sin a T^ 

^ tan Ad . tan q> , tan Ad . tan d ^ 

sm aTI = : ^ H cos a Jl 

sin ^ tan t 

This is a rigorous expression of the required correction a 7^, but 
the change of declination is so small that we may put a5 for its 
tangent, a 7^ for its sine, and unity for cos aTJ^, without any 
appreciable error ; and, since a5 is expressed in seconds of arc, 
we shall obtain aT^ in seconds of time by dividing the second 
member by 15. We thus find the formula* 

aT = — ^^- ^^"_? . ^^'^rid 

" 15 8in< "^ 15tan< ^ ^ 

The Ephemeris gives the hourly change of d. If we take it for 
the Greenwich instant corresponding to the local noon, and call 
it A 'dj and if t is reduced to hours, we have 

Ad = A'd . ^ 

and our formula becomes 

aT.^: — 

A^d.ftany aM . f tan d pEquationl ^263^ 
15 sin t 15 tan t Lfor noonJ 

To facilitate the computation in practice, we put 

A = ^~ — — Bz= 

15 sin t 15 tan t 

a=^.A'd.tanf 6 = J?. A'd.tand ) (264) 

then we have 

* As first giyen by Gauss, Monatliche Corretpondenz^ Vol. 28. 

200 TIME. 

The correction aT^ is called the equation of equal altitudes. The 
computation according to the above form is rendered extremely 
simple by the aid of our Table IV., which gives the values of 
log A and log J5 with the argument "elapsed time" (=20. 
Then a and b are computed as above, the algebraic signs of the 
several factors being duly observed. When the sun is mo\'ing 
towards the 7iorthy give a'<J the positive sign ; and also when 
^ and 3 are north, give them the positive sign ; in the opposite 
cases they take the negative sign. The signs of A and J5 are 
given in the table ; A being negative only when t < 12* and B 
positive when ^ < 6* or > 18*. 

When we have applied ^T^to the mean of the clock times (or 
the "middle time"), we have the time 

as shown by the clock at the instant of the sun's meridian transit 
Then, computing the time T', whether mean or sidereal, which 
the clock is required to show at that instant, we have the clock 
correction, as before, 

at= r^ t 

Example. — March 6, 1856, at the U. S. Naval Academy, Lat 
38° 59' N., Long. 5* 5"* 57\5 W., the sun was observed at the 
same altitude, A.M. and P.M., by a chronometer regulated to 
mean Greenwich time ; the mean of the A.M. times was 1* S" 26'.6, 
and of the P.M. times 8* 45"" 4r.7 ; find the chronometer cor- 
rection at noon. 

Wo have first A.M. Chro. Time = 1* 8- 2(>'.G 

P.M. '' *' =8 45417 
Elapsed time 2< =7 37 15.1 

Middle time T; = 4 57 4 .15 

From the Ephemeris we find for the local apparent noon of 
March 5, 1856, 

^ :r^ — 5° 46' 22".5 Equation of time = + 11- 35M1 
A'a = + 58'MO 

For the utmost precision, we reduce ^'d to the instant of local 


noon. With these quantities and ^ = 38® 59', we proceed as 
follows : 

Arg. 7» 37- Table IV. log A n9.4804 log B 9.2151 

logA'^ 1.7642 logA'^ 1.7642 
log tan ^ 9.9081 log tan d n9.0047 

log a nl.l527 log 6 n9.9840 
a = — 14'.21 b = — 0*.96 

Middle Chro. time T^ = 4» 67- 4'.15 

^T^ = a + b = — 15.17 

Chro. Time of app. noon T = 4 56 48.98 

This quantity is to be compared with the Greenwich time of the 
local apparent noon, since the chronometer is regulated to 
Greenwich time. We have 

Mean local time of app. noon = 0* 11* 35*.ll 

Longitude =5 5 57 .50 

Mean Greenwich time " = T' = 5 17 32.6 1 

A r = r' — T = + 20- 43'.63 

If the correction of the chronometer to mean local time is 
required, we have only to omit the application of the longitude. 
Thus, we should have 

Chro. time of app. noon = 4* 56* 48'.98 
Equation of time = — 11 35 .11 

Chro. time of mean noon = 4 45 13 .87 

and since at mean noon a chronometer regulated to the local 
time should give 0* 0** 0*, it is here fccstj and its correction to 
local time is — 4* 45"* 13'.87. 

141. (C.) Efjual altitudes of the sun in the afternoon of one day and 
the morning of the next folloxcing day ; i.e, hrforc and after midnight, — 
It is evident that when equal zenith distances are observed in 
the latitude + ^, their supplement to 180° may be considered as 
equal zenith distances ob8er\"ed at the antipode in latitude — (p 
on the same meridian. Hence the formula (263) will give the 
equation for noon at the antipode by substituting — <f for + ^, 
that is, by changing the sign of the first term ; but this noon at 

202 TIME. 

the antipode is the same absolute instant as the midnight of the 
observer, and hence 

£^T = ^'^ ' ^ ^^^ ^ -t- ^'^-^^^"^ pquation fori ,^^. 
° 15 sin ( 15 tan t L midnight. J 

and this is computed with the aid of the logarithms of A and B 
in Table IV. precisely as in (264), only changing the sign of A. 
The sign for this case is given in the table.* 

142. To fold the correction for small inequalities in the altitudes. — 
If from a change in the condition of the atmosphere the re- 
fraction is different at the two observations, equal apparent alti- 
tudes will not give equal true altitudes. To find the change ^t 
in the hour angle t produced by a change aA in the altitude A, 
we have only to difterentiate the equation 

sin A =: sin ^ sin d -f cos 9 cos d cos t 

regarding (p and d as constant ; whence 

cos h , ^h =z — cos tp cos d sin t . \b^t 

where aA is in seconds of arc and ^t in seconds of time. 

If the altitude at the icest observation is the greater by aA, the 
hour angle is increased by a^ and the middle time is increased by 
J a/. The correction for the difference of altitudes is therefore 
— \ a/, and, denoting it by a' 7^y, we have, by the above equation, 

A' 7;=. ^'^-•-"'*^* (266) 

30 cos ff cos ^ sin ^ 

This correction is to be added algebraically to the middle clock 
time in any of the cases (A), (13), (C) of the preceding articles. 

Example. — Suppose that in Example 2, Art. 139, there had 
been observed at the east observation Barom. 30.30 inches. 
Therm. 35° F., but at the west observation Barom. 29.5/) inches, 
Therm. 52° F. AVe have for the altitude 52° 5' or zenith dis- 
tance 37° 55', bv Table I., the mean refraction 45".4. By Table 

* For an example and some practical remarks, nee my *' lmprove4l meili(Hl of 
fin»liii}jj tlie error ami rate of a chronometer by equal altituder*/' Appendix lo the 
American Kpliemeris for 185C and 1857. 




log aA 



log cos h 



log 80C 9 



log sec d 



log cosec i 

t 0.270 

iog 3 





XIV.A and XiV.B, the corrections for the barometer and ther- 
mometer are as follows, taking for greater accuracy one-eighth 
of the corrections for 6' : 

East Obs. West Obs. 

Barom. 30.30 + 0".5 Barom. 29.55 — 0".6 

Therm. 35°. + 1 .4 Therm. 52°. —0.1 

+ 1 .9 — .7 

The difference of these numbers gives aA = + 2".6 as the excess 

of the true altitude at the west observation. Hence, by the 

formula (266), 

A/i = + 2".G 

h= 52^ 

ip = — 33 

a = — 10 

t = \ elapsed time = 2* 9* 

a'7;=+ 0M2 

When, however, several altitudes have been observed, as in 
this example, we may obtain this correction from the observa- 
tions themselves ; for we see that the double altitude of Spica 
changed 20' = 1200" in about 55*, and hence we have the 


1200" : 2".6 = 55* : a' T; 

which gives aTo = + 0*.12 as before. By taking the change in 
the double altitude, the fourth tenn is the value of Ja^, or a'TJ,. 

If this correction be applied, we lind the corrected time of 
transit = 12* 30"* 19*.12, and consequently the chronometer cor- 
rection Ar= — 1"* 45*.54. 

The altitudes may difter from other causes besides a change in 
the refraction ; for instance, the second observation may be in- 
termitted by passing clouds, so that the precisely corresponding 
altitude cannot be taken ; but, rather than lose the whole ob- 
servation, if we can observe an altitude diftcring but little from 
the first, we may use it as an equal altitude, and compute the 
correction for the difterence by the formula (260). 

143. Effect of oTors in the latituik^ (Iccllnaiion, and aliltnde upon 
the time found by equal altitudes. — The time found by e([ual altitudes 
of a fixed star is wholly independent of errors in the latitude 

204 TIME. 

and declination, since these quantities do not enter into the com- 
putation. In observations of the sun, an error in the latitude 
afiects the term 

a =^ A^'d tan f 

by differentiating which we lind that an error d(p produces in a 
the error da = A a'5 . sec^ f . df^ or, putting sin dtp for rf^, 

da = A ^'d see' ^ sin df 

In the same manner, we find that an error dd in the declination 
produces in b the error 

db = B^'d see' d sin dd 

In the example of Art. 140, suppose the latitude and declina- 
tion were each in error V. We have 

logAr^'d nl.2446 log B^'S 0.9793 

logsec'v^ 0.2188 log sec« ^ 0.0044 

log sin 1' _6.4637 log sin 1' 6.4687 

log da n7.9271 logdb 7.4474 

da=— 0'.008 db=-\- 0*.003 

If dtp and dd had opposite signs, the whole error in this case would 
be 0*.008 + 0\003 = O'.Oll. As the obsci-ver can always easily 
obtain his latitude within 1' and the declination (even when the 
longitude is somowhat uncertain) within a few seconds, we may 
regard the method as practically free from the effects of any 
errors in these quantities. The accuracy of the result will there- 
fore depend wholly ui)on the accuracy of tlie observations. 

The accuracy of the observations dci>ends in a measure upon 
the constancy of the instrument, but chiefly upon the skill of the 
observer. Each observer may determine the probable error of 
his observations by discussing them by the method of least 
squares. An example of such a discussion will be given in the 
foUowiuic article. 

The effect of an error in the altitude is given by (266). Since 
we have, A being the azimuth of the object, 

cos ^ sin t 
sm A = 

cos h 


the formula may also be written 

A'7; = ^ 

30 cos ^ sin A 

which will be least when the denominator is greatest, i.e. when 
A = 90° or 270°, or when the object is near the prime vertical. 
From this we deduce the practical precept to take the observations 
whai the object is nearly east or loest. This rule, however, must not 
be carried so far as to include observations at very low altitudes, 
where anomalies in the refraction may produce unknown dif- 
ferences in the altitudes. If the star's declination is very nearly 
equal to the latitude, it will be in the prime vertical only when 
quite near to the meridian, and then both observations may be 
obtained within a brief interval of time ; and this circumstance 
is favorable to accuracy, inasmuch as the instrument will be less 
liable to changes in this short time. 

144. Probable error of observation. — The error of observation is 
composed of two errors, one arising from imperfect setting of 
the index of the sextant, the other from imperfect noting of the 
time ; but these are inseparable, and can only be discussed as a 
single error in the observed time. The individual observations 
are also aftected by any irregularity of graduation of the sextant, 
but this error does not aftect the mean of a pair of obser\'ations 
on opposite sides of the meridian ; and therefore the error of 
observation proper will be shown by comparing the mean of 
the several pairs with the mean of these means. K, then, the 
mean of a pair of observed times be called a, the mean of all 
these means a^,, the probable error of a single pair, supposing all 
to be of the same weight, is* 



in which n = the number of pairs, and q = 0.6745 is the factor 
to reduce mean to probable errors. The probable error of the 
final mean a^ is 

r„ = 


* See Appendix I Leatt Squares. 

206 TIME. 

Example. — At the U. S. Naval Academy, June 18, 1849, the 
following series of equal altitudes of the sun was observed. 

Chro. A.M. 




a — Oo 


0» 43^ 53'. 


' 3'. 5 

5* 13"» 58'.25 

-f 0'.12 


44 19. 




+ 0.37 


44 45. 






45 11. 




+ 0.52 


45 37. 




+ 0.22 


46 1.7 




— 0.58 


46 28.5 



57 .75 



46 55. 






47 19.7 





— 0.08 






» = 9 

r — 


A similar discussion of a number of sets of equal altitudes of 
the sun taken by the same observer gave 0'.23 as the probable 
error of a single pair for that observer, and consequently the 
probable error of the result of six observations on each side of 
the meridian would be only 0'.23 -- 1/ 6 = 0\094. This, liow- 
ever, expresses only the accidental error of obsaTation^ and does 
not include the eftcct of changes in the state of the sextant be- 
tween the morning and afternoon observations. Such changes 
are not unfrcqncntly produced by the changes of temperature to 
which it is exposed in obsen'ations of the sun; it is important, 
therefore, to guard the instrument from tiie sun's rays as much 
as possible, and to exjjose it only during the few minutes 
required for each observation. The determination of the time 
by stars is mostly free from difficulties of this kind, but the 
observation is not otherwise so accurate as that of the sun, ex- 
cept in the hands of very skilful observers. 


145. Let the altitude of anv celestial bodv be observed with 

»' • 

the sextant or anv altitude instninieut, aud the time noted bv 
tlie clock. For greater precision, observe several altitudes in 
([uick succession, noting the time of each, and take the mean of 
the altitudes as to the mean of the times. But 


hi taking the mean of several observations in this way, it mnst 
not be forgotten that we assume that the altitude varies in pro- 
portion to the time, which is theoretically true only in the 
exceptional ease where the observer is on the equator and the 
star's declination is zero. It is, however, practically tnie for an 
interval of a few minutes when the star is not too near the 
meridian. The obser\'ations themselves will generally show the 
limit beyond which it will not be safe to apply this rule. When 
the observations have been extended beyond this limit, a cor- 
rection for the unequal change in altitude (i.e. for second diifer- 
ences) can be applied, which will be treated of below. 

With the altitude and azimuth instrument we generally ob- 
tain zenith distances directly. In all cases, however, we may 
suppose the observation to give the zenith distance. Having 
then corrected the observation for instrumental errors, for re- 
fraction, &c.. Arts. 135, 136, let ^ be the resulting true or geo- 
centric zenith distance. Let (p be the latitude of the place of 
observation, d the star's declination, / the star's hour angle. 
The three sides of the spherical triangle formed by the zenith, 
the pole, and the star may be denoted by a = 90° — ^, 6 = ^, e = 
90° — 5, and the angle at the pole by 13 = ^, and hence. Art. 22, 
we deduce 

,inj,_ /( Bi»K: + (y -3)] sin K:-(y -«>)] ) (207) 

\ \ cos yj cos d I 

which gives / by a very simple logarithmic computation. From 
/ we deduce, by Art. 55, the local time, which compared with 
the ob3er\'ed clock time gives the clock correction required. 

It is to be obsei^ved that the double sign belonging to the 
radical in (267) gives t\vo values of sin J /, the positive corre- 
sponding to a west and the negative to an east hour angle; since 
any given zenith distance may be observed on either side of the 
meridian. To distinguish the true solution, the observer must 
of course note on which side of the meridian he has observed. 

If the object ob8er\x»d is the sun, the moon, or a planet, its 
declination is to be taken from the Ephemeris, for the time of 
the obser\'ation (referred to the meridian of the Ephemeris); but, 
as this time is itself to be found from the observation, we must 
at first assume an approximate value of it, with which an approxi- 
mate declination is found. With this declinatibn a first compu- 

208 TIME. 

tation by the formula gives an approximate value of t, and hence 
a more accurate value of the time, and a new value of the decli- 
nation, with which a second computation by the formula gives a 
still more accurate value of L Thus it appears that the solution 
of our problem is really indirect, and theoretically involves an 
infinite series of successive approximations; in practice, how- 
ever, the observer generally possesses a sufficiently precise value 
of his clock correction for the purpose of taking out the declina- 
tion of the sun or planets. The moon is never employed for 
determinhig the local time except at sea, and when no other 
object is available.* 

Example. — At the U. S. Naval Academy, in Latitude <p = 38° 
68' 53" N., Longitude 5* 5- 57'.5 W., December 9, 1851, the fol- 
lowing double altitudes of the sun west of the meridian were 
observed with a sextant and artificial horizon, the times being 
noted by a Greenwich mean time chronometer: 



7* 35- 14'.5 

SS*' 30' 

Barom. 30.28 inches. 

35 55. 

« 20 

Att. Therm. 55** F. 

36 35.5 

" 10 

Ext. Therm. 50^ F. 

37 15.5 


Index correction of the 

37 55. 

32 50 

soxtaut — 1' 10" 

Means 7 36 35.1 33 10 

The approximate correction of the chronometer was assumed to 
be + 9"* 40*. Find its true correction. 

With the assumed chronometer correction we obtain the aj> 
proximate Greenwich time = 7* 46"* 15% with which we take 
from the Ephemeris 

d=.-. — 22° 50' 27" Sun's semidiametcr S = 16' 17" 
Eq. of time :^ — 7" 25'.80 " hor. parallax r = 8".7 

"We have then 

♦ But the moon's altitude and the hour nngle deduced from it may be used in 
finding the observer's longitude, as will be shown in the Chapter on Longitude. 

t The -ymbol Q is used for "observed altitude of the sun's lower limb/* and 2Q 
for the double altitude from the artificial horizon. In a similar manner we um 

Cy» X 'j>' 


Observed 2 © =83^ 10' 0" 
Index corr. = — 1 10 

33 8 50 

App. altitude = 16 34 25 

^r = 73 25 35 

(Table II.) r = + 3 15 

7c Gin z = p =2 — 8 

/S= — 16 17 

C = 73 12 25 

The computation by (267) is then as follows : 

sr= 38^58' 53" log sec s«^ 0.109383 

^ = — 22 50 27 log sec d 0.035464 

^ — ^ = 61 49 20 log sin } sum 9.965661 

C = 73 12 25 log sin } diff. 8.996455 

i sum = 67 30 52 .5 19106963 

J diflf. = 5 41 32 .5 log sin } f 9.553482 

J f = 20« 57' 25".6 
Apparent time = ^ = 2* 47"» 39'.4 
Eq. of time = —7 25.8 

Local mean time = 2 40 13 .6 
Longitude =5 5 57 .5 

True Gr. Time =T'= 7 46 11.1 

T= 7 36 35.1 

A!r=+ 9 36.0 

agreeing so nearly with the assumed correction that a repetition 
of the computation is unnecessary. 

146. If it is preferred to use the altitude instead of the zenith 
distance, put the true altitude h = 90° — ^, and the polar distance 
of the star P = 90° — d, then we have, in (267), 

sini[C— (f — ^)]=8inJ(90° — A — ^ + 90°— P) = cosJ(A+^ + P) 
siniCC+f' — ^]=sinj(90°— A + sp — 90^ + P) = 8m J(^ + P^A) 

If then we put 

the formula becomes 


Vol. L— 14 

. ,^ //C08 5 sin (5 -- A)\ 

\\ cosy>sinP / ^ ' 

210 TIME. 

In this form we may always take P = tlie distance from the ele- 
vated pole, and regard the latitude as always positive, and then 
no attention to the algebraic signs of the quantities in the second 
member is required. Thus, in the preceding example, we should 
proceed as follows : . 

App.alt.= 16^ 34' 25" 
r— ;>= — 3 7 
S= 16 17 

A = 16 47 35 

^= SS 5S 53 log sec 0.109383 

P = 112 50 27 log cosec 0.035464 

25 = 168 36 55 

8= 84 18 27 .5 log cos 8.996455 

5 — A = 67 30 52 .5 log sin 9.965661 

and the computation is finished as in the preceding article. 

147. If we aim at the greatest degree of precision which the 
logarithmic tables can afford, we should find the angle J/ by its 
tangent, since the logarithms of the tangent always vary more 
rapidly than those of the other functions. For this purpose wo 

tanit = Jl Bi!L(^^^)^^n.(.-a) ^ } (269) 

' \ cos S COS (s — C) / 

or, if the altitude is used, 

s = 1 (A + y + -P) ) 

^ \ sin (s — ^) cos {s — P) / 

148. If a number of observations of the same star at the same 
place arc to be individually computed, it will be most readily 
done by the fundamental equation 

cos C — Bin <p sin d 
cos t = 

cos ^ COB d 


for the logarithms of sin y> sin S and cos <p cos 5 will be constant, 
and for eacli observation we shall only have to take from the 
trigonometric table the log. of cos ^ ; the logarithm of the nume- 
rator ^\l]\ then be found by the aid of Zecii's Addition or Sub- 
traction Table, which is included in IIulsse*s edition of Vega's 
Tables. The addition or the subtraction table vnll be used ac- 
cording as sin y> sin d is positive or negative. 

149. JEffcci of errors in the data upon the time computed from an 
altitude. — We have from the dilterential equation (51), Art. 35, 
multiplymg dt by 15 to reduce it to seconds of arc, 

sin q cos d (15 dt) = d^ — cos A d<p -j- cos q dd 

wliere e/^, dip, d8, may denote small errors of ^, y>, d, and dt the 
corresponding error of t;Ais the star's azimuth, q the parallactic 
angle, or angle at the star. 

If the zenith distance alone is erroneous, we have, by putting 
df = 0, and dd = 0, 

Udt= "^^ ^^ 

sin q cos d cos ^ sin A 

from which it follows that a given error in the zenith distance 
will have the least eftect upon the computed time when the 
azimuth is 90^ or 270° ; that is, when the star is on the prime 
vertical; for we then have m\A = ± 1, and the denominator 
of this expression obtains its maximum numerical value. Also, 
since cos y> is a maximum for f> = 0, it follows that observa- 
tions of zenith distances for determining the time give the 
most accurate results when the place is on the equator. On the 
other hand, the least favorable position of the star is when it is 
on the meridian, and the least favorable position of the observer 
is at the pole. 
By putting d!^ = 0yd8 = 0, sin q cos 8 = cos y> sin A we have 

ibdt=- ^'^ 

cos ip tan A 

by which we see that an error in the latitude also produces the 
least eftect when the star is on the prime vertical, or when the 
obser^-er is on the equator. Indeed, when the star is exactly in 

212 TIME. 

tlic prime vertical, a small error in ip has no appreciable effect : 
since, then, tau^l =^ ac, and hence when the latitude is uncertain, 
we may still obtain good results by obser\'ing only stars near the 
prime vertical. 

By putting rf^ = 0, d(p = 0, we have 

cos J tan (I 

which shows that the error in the declination of a given star 
produces tlie least effect when the star is on the prime vertical ;* 
and of different stars the most eligible is that which is nearest 
to the equator. 

As verj' great zenith distances (greater than 80°) are, if jws- 
sible, to be avoided on account of the uncertainty in the refraction, 
the observer will often be obliged, especially in high latitudes, 
to take his obsen'ations at some distance from the prime vertical, 
in which case snuill erroi's of zenith distance, latitude, or declina- 
tion may have an important effect upon the computed chxh ror- 
revtion. Nevertheless, constant errors in these quantities will 
have no sensible effect upon the rate of the clock deduced from 
zenith distances of the same star on different davs, if the star is 
observed at the same or nearly the same azimuth, on the same 
side of the meridian ; for all the clock corrections will be in- 
creased or diminished by the same quantities, so that their 
differences, and consequently the rate, will be the same as if 
these errors did not exist. The errors of eccentricitv and 
graduation of the instrument are among the constant errors 
which niav thus be eliminated. 


But if the same star is observed both east and west of the 
meridian, and at the same distance from it, sin .4 or tan^l, and 
tan 7, will be positive at one observation and negative at the 
other, and, having the same numerical value, constant errors 
dif, (ft), and (l;^ will give the same numerical value of (ft with 
op[»ositc signs. Hence, while one of the deduced clock correc- 
tions will be too great, the other will be too small, and their 
mean will be the true correction at the time of the star's transit 

* From the equation sin g = — sin Ay it follows that sin ^ is a maximum 

cos (S 

(for constant values of o and 6) when sin .1 ::= 1, and tan ^ is a maximam in the 

enme case. 


over the meridian. Ilenee, it follows again, as in Art. 143, that 
small errors in the latitude and declination have no sensible 
cflect upon the time computed from equal altitudes. 

150. To find the change of zenith distance of a star in a given in- 
terval of time^ having regard to second differences. 
The formula 

dZ = COS ^ sin A dt 

is strictly true only when rfj and dt are infinitesimals. But the 
complete expression of the finite difference a^ in terms of the 
finite diflference ^t involves the square and higher powers of ^t. 
Let ^ be expressed as a function of t of the form 


then, to find any zenith distance ^ + ^C corresponding to the 
hour angle t + a^, we have, by Taylor's Theorem, 

c + ^:=f(t + An =ft + -^'^t+—^- h--- 

^ dt dt^ 2 

or, taking only second differences, 

^ d: ^ , dK A<' 

AC = A^ H 

dt dt' 2 

We have already found 


= cos f> sin A 


which gives, since A varies with t, but <p is constant, 

dK , dA 

— = cos cos A • — 

dt* dt 

But from the second of equations (51) we have, since dd and dtp 

are here zero, 

dA cos q cos d cos q sin A 

dt sin C sin t 


d*^ cos ^ sin A cos A cos g 

dt* smt 

214 TIME. 

and the expression for a^ becomes 

. - ^ , cos cp sin^ cosul cosflf ^t^ 
AC = cos ^BmA.^t-\ ^ 

sin t 2 

Since a^ and a^ are here supposed to be expressed in parts of 
the radius, if we wish to express them in seconds of arc and of 
time respectively, we must substitute for them a^ sin V and 
15 A< shi 1", and the formula becomes 

• A ^ir ^N , COR v» sin il cos -4 cos flf flSAO'sinl" ,^.,, 

AC = cos ^ sin A (15 a^ H • ^^ (271) 

sin t 2 

But in so small a term as the last we may put 

(15A0'sin 1" 2sin» jAf 
2 sin 1" 

the value of which is given in our Table V., and its logarithm 
in Table VI. ; so that if we put also 

. , , cos A cos a 
a = cos ^ sm-4, A* = ^ 

sin t 
we shall have 

aC = 15aA^ + akm (272) 

151. A number of zenith disianees being observed at given o/ocJt 
times, to correct the mean of the zenith distances or of the clock time^ 
for second differences, — The first term of the above value of a^ 
varioH in j^roportiou to a^, but the second term varies in projwr- 
tion to a/^; and hence, when the interval is sufficiently great to 
render this second term sensible, equal intervals of time corre- 
spond to unequal difterences of zenith distance, and vice versa: 
in otlier words, we shall have second diflerences either of the 
zenith distance or of the time. Two methods of correction 
present themselves. 

1st. Reduction of the mean of the zenith distances to the mean of the 
times. — Let 7'„ T^, T^, &c. be the observed clock times ; ^^, ^^ ^j, 
&c. the corresponding obsei'ved zenith distances; 7* the mean ot 
the times; i^o ^^^^ mean of the zenith distances; ^ the zenith 
distance corresponding to 71 The change Ci ~ C eorresjwnds to 
the interval 7; — 7\ Ca — C ^<^ ^2 — ^\ ^^' ; »« that if we put 

7;- T=T,, T,- r = r.,&c. 


we have, by (272), 

Ci — C = 15 a Tj -|- aknii 
C, — C = 15 a T, + akm^ 
C, — C = 15 a T, + akm^ 
&c. &c. 

m which Tiu = — : — — — % rru = — : — —-5, &c., are found by Tab. V. 
^ sin 1" ' ^ sin 1" ' ' -^ 

with the arguments Tj, r„ &c. The mean of these equations, 
observing that 

■^1 + ^9 + T. + Ac. = 

C=C g/; ^ + ^, + Wl, + &C 

" n 

in which n = the number of observations. Or, denoting the mean 
of the values of m from the table by ?w^„ that is, putting 

wi, -|- m, + m, + &c. 

• n 

we have 

C = Co - akm, (278) 

2rf. Reduction of the mean of the times to the mean of tlie zenith 
distances. — Let T^ be the clock time corresponding to the mean 
of the zenith distances, then Za — C i® ^^ change of zenith dis- 
tance in the interval T^ — jT, and, since this interval is very small, 
we shall have sensibly 

15a (j; — T) = C. — C = akm^ 

T,= T + -r^km, (274) 

We have, then, only to compute the true time 7^/ from the mean 
of the zenith distances in the usual manner, and the clock cor- 
rection will then be found, as in other cases, by the formula 

To compute A-, we must either first find q and A^ or, which is 
preferable, express it by the known quantities. We have 

cos gr cos^ = cos t — sin jr sin A cos C 

== cos t cos ip cos ^ cos C 


216 TIME. 


m mil X a 1 ^^^^ COS f> COS d ,o^cv 

r, = r + J5 m, cot t - Jy ^« — . ^/ ^ (275) 

sin C tan C 

in which we employ for ^ and < the mean zenith distance and 
the computed hour angle. 

This mode of correction is evidently more simple and direct 
than the first. 

Example.— In St. Louis, Lat. 38° 38' 15" K, Long. 6* 1" 7* W., 
tne following double altitudes of the sun w^ere observed with a 
Pistor and Martin prismatic sextant, the index correction of 
which was + 20". The assumed correction of the chronometer 
to mean local time was + 2*^ 12*. Barom. 30.25 inches, Att 
Therm. 80°, Ext. Therm. 81°. 

St. Louis, June 24, 1861. 


1250 16' 10". 

125 49 10 

12G 23 

12G 41 40 

127 32 80 

127 67 45 

128 22 

128 61 50 

129 8 35 

129 38 


127 33 28 

-]- 20 

127 33 48 


63 4G 54 


— 27 .2 

P — 

-1- 3 .7 

S = 

-1- 15 40 .3 


64 2 16 .8 

Co = 

25 57 43 .2 

^ - 

38 38 15 . 

(5 = 

23 23 49 .3 

t - 

24° 43' 48".4 

— . 


App. time — 

22 21 4.77 

Eq. of time ^^ 

4- 2 18.17 

7" - 

^0 - 

22 23 22.94 




22* 14*" 









50 .73 





28 .14 





12 .76 









2 .67 










28 .60 





42 .46 





67 .19 

22 21 



lo — 82 .66 

Correction for 1 ^ «,. 

second diff. /JI L1 ^^«^o 1-^139 

T --22 21 10.48 ^^* ^ 8.8239 

^/ __ 22 03 00 94 ^^K ^^^ ^ «0.3367 

AT-:- + 2 12.46 

4*. 73 wO.6745 

log ^ m, 0.3878 

log sin t n9.6215 

log cos ^ 9.8927 

log cos d 9.9627 
log cosec :^ 0.8688 

log cot Z^ 0.8125 

— 8*.06 fiO.4a60 

— 1.67 

*Thc refraction should here be the mean of the refractions computed for the 


The correction for second difterences is particularly useful in 
reducing series of altitudes observed with the repeating circle ;* 
for with this instrument we do not obtain the several altitudes, 
but only their mean. (See Vol. 11.) Wlien the several altitudes 
are knowni, we can avoid the correction by computing each 
observ^ation, or by dividing the whole series into groups of such 
extent that ^\'ithin the limits of each the second difterences will 
be insensible, and computing the time from the mean of each 



152. The rate of the clock may be found by this method with 
considerable accuracy without the aid of astronomical instru- 
ments. The terrestrial object should have a sharply defined 
vertical edge, behind which the disaj^i^earance is to be obser\'ed, 
and the position of the eye of the observer should be j^recisely 
the same at all the observations. If the star's right ascension 
and declination are constant, the difterence between the sidereal 
clock times 7^ and T^ of two disappearances is the rate dTm the 
interval, or 

but if the right ascension a has increased in the interval by Aa, 
then the rate is 

dT=T^— T^ + Aa 

To find the correction for a small change of declination = a5, 

several altitudes or zenith distances, but for small zenith distances the difference 
will be insensible. At great zenith distances we should compute the several refrac- 
tions, but under 80^ we may take the refraction r for the mean apparent zenith 
distance z^, and correct it as follows : Take the difference between z^ and each z, and 
the mean m^ of the values of 

^_ 28inM(^-^o) 
sin \" 

from Table V. (converting the argument z — z^ into time) ; then the mean of the 
refractions will be found by the formula 

r^ = r -j- 2»Iq sin r sec* i^ 

The difference z — z^ should not much exceed 1°. 

* This method was frequently practised in the geodetic survey of France. See 
Nouvelle Deseriplion GSomSlrique de la France (Puisbamt), Vol. I. p. 96. 

218 TIME. 

we have, by the second equation of (51), since the azimuth is here 
constant as well as the latitude, so that dA = and df = 0, 

a5 tan a 
15 cos d 

and hence the rate in the interval will be 

iT=T,-T, + ^a-^^^^ (276) 

15 cos d 

The angle q will be found with sufficient precision from an 
approximate value of i by (19) or (20). 

If we know the absolute azimuth of the object, we can find 
the hour angle by Art. 12, and hence also the clock correction. 


153. To fold the time of true rising or setting, — that is, tlie instant 
when the star is in the true horizon, — ^we have only to compute 
the hour angle by the formula (28) 

cos t =z — tan <p tan d 
and then deduce the local time by Art. 55. 

154. To find the time of apparent rising or setting, — that is, the 
instant when the star appears on the horizon of the observer, — we 
must allow for the horizontal refraction. Denoting this refraction 
by r^, the true zenith distance of the star at the time of apparent 
rising or setting is 90° + r^, and, employing this value for ^, we 
compute the hour angle by (2G7). 

Since the altitude A = 90° — (J^, we have in this case h = — r^ 
with which we can compute the hour angle by the formula (268). 

In common life, bv the time of sunrise or sunset is meant the 
instant when the sun's upper limb appears in the horizon. The 
true zenith distance of the centre is, then, ^ = 90° -\- r^ — r -•- S 
(where z - the horizontal parallax and *V = the semidiameter), 
with which we compute the hour angle as before. The same 
form is to be used for the moon. 


155. Twilight begins in the morning or ends in the evening 
when the sun is 18° below the horizon, and consequently the 

AT SEA. 219 

zenith distance is then ^ = 90° + 18°, or A = — 18°, with which 
we can find the hour angle by (267) or (268). 

NoTK. — Methods of finding at once both the time and the latitude from observed 
altitudes will be treated of under Latitude, in the next chapter. 


First Method. — By a Single Aliiiudc. 

156. This is the most common method among navigators, as 
altitudes from the sea horizon are observed with the greatest 
facility with the sextant. Denoting the observed altitude cor- 
rected for the index error of the sextant by H^ the dip of the 
horizon by X), we have the apparent altitude h' = H — D; then, 
taking the refraction r for the argument A', the true altitude of a 
star is A = A' — r. A i:)lanet is observed by bringing the esti- 
mated centre of its reflected image upon the horizon, so that no 
correction for the semidiameter is employed; the jjarallax is com- 
puted by the simple formula (;r being the horizontal parallax) 

p = 7: cos A' 

and hence for a planet 

h = h' — r -{- IT cos A' 

The moon and sun are observed by bringing the reflected 
image of either the upper or the lower limb to touch the horizon. 
As very great precision is neither possible nor necessary in these 
observ^ations, the compression of the earth is neglected, and the 
parallax is computed by the formula 

p z= n cos (A' — r) 
and then, S being the semidiameter, 

A = A' — r + TT cos (A' — r) ± S 

In nautical works, the whole correction of the moon's altitude 
for parallax and refraction = ;: cos (A' — r) — r is given in a table 
with the arguments apparent altitude (A') and horizontal jiarallax 
(r). In the construction of this table the mean refraction is used, 
but the corrections for the barometer and thermometer are given 
in a very simple table, although they are not usually of suflicient 
importance to be regarded in correcting altitudes of the moon 
which are taken to determine the local time. 

220 TIME. 

The hour angle is usually found by (268). 

It is important at sea, where the latitude is always in some 
degree uncertain, to iind the time by altitudes near tlie prime 
vertical, where the error of latitude has little or no eftect 
(Art. 149). 

157. The instant when the sun's limb touches the sea horizon 
may be observed, instead of measuring an altitude with tlie sex- 
tant. In this case the refraction should be taken for tlie zenith 
distance 90° + Z), but, on account of the uncertainty in the liori- 
zontal refraetion, great precision is not to be expected, and the 
mean horizontal refraction i\ may be used. We then have 
l^ == QO"" + D + r^^— t: ± Sy with which we proceed by (267). In 
so rude a method, t: may be neglected, and we may take 16' as 
the mean value of Sj 36' as the value of r^,, 4' as tlie average 
value of/) from the deck of most vessels; then for the lower 
limb we have C = 90° 56', and for the upper limb r = 90° 24'. If 
both limbs have been observed and the mean of the times is 
taken, the corresponding hour angle will be found by taking 
C = 90° 40'. 

Second Method. — By Equal Altitudes. 

158. The method of equal altitudes as explained in Arts. 139 
and 140 may be applied at sea by introducing a correction for 
the ship's change of place between the two observations. If, 
however, the ship sails due east or west between the obsen'a- 
tions, and thus without changing her latitude, no correction tor 
her change of place is necessary, for the middle time will evi- 
dently correspond to the instant of transit of the star over the 
middle meridian between the two meridians on which the equal 
altitudes are observed. But, if the ship changes her latitude, 

Af = the increase of latitude at the second observation; 
then (Art. 149) the eftect upon the second hour angle is 

Af r=: — 

15 cos <p tan A 

which is the correction subtractive from the second obBorved 
time to reduce it to that which would have been observed if the 

AT SEA. 221 

ship had not changed her latitude or had run upon a parallel. 
Hence Ja< is to be subtracted from the mean of the chrono- 
meter times to obtain the chronometer time of the star's transit 
over the middle meridian. 

In this formula we must observe the sign of tan A, It will 
be more convenient in practice to disregard the signs, and to 
apply the numerical value of the correction to the middle time 
according to the following simple rule : — add the correction when 
the ship has receded from the sun; subtract it when the ship has 
approached the sun. 

The azimuth may be found by the formula 

. sin t cos d 

sm A = 

cos h 

in which for t we take one-half the elapsed time. 

The sun being the only object which is employed in this way, 
we should also apply the equation of equal altitudes, Art. 140; 
but, as the greatest change of the sun's declination in one hour 
is about 1', and the change of the ship's latitude is generally 
much greater, the equation is commonly neglected as relatively 
unimportant in a method which at sea is necessarily but ap- 
proximate. But, if required, the equation may be computed 
and applied precisely as if the ship had been at rest. 

Example. — At sea, March 20, 1856, the latitude at noon being 
39° N., the same altitude was obser\^ed A.M. and P.M. as fol- 
lows, by a chronometer regulated to mean Greenwich time : 

Obsd-X^ 30° 0' A.M. Chro. time = 11* 39-» 33 

Index corr. — 2 P.M. *' " = 6 20 17 

Dip — 4 Elapsed time = 2^ = 6 40 44 

Refraction — 2 Middle time = 2 59 55 

Semidiam. + 16 Chron. correction = — 2 12 

A = 30 8 Green, time of) 9 57 4^ 

noon J 

The ship changed her latitude between the two observations 
by A^ = — 20' = — 1200". For the Greenwich date March 
20, 2* 58'", the Ephemeris gives 5 = + 0° 4', and we have t = 
8* 20« 22- = 50° 5' 30'', ip = 39° 0'. Hence 

222 TIME. 

log sin t 9.8848 log j'^ 8.5229 

log cos d 0.0000 log A^ 3.0792 

log sec h 0.0631 log sec <p 0.1095 

log sin A 9.9479 log cot A 9.7165 

log26'.8 1.4281 

The ship has approached the sun, and hence 26*.8 must be sub- 
tracted from the middle time. 

K we wish to apply the equation of equal altitudes, we have 
further from the Ephemeris £^'d = + 59", and hence, by Art 


log A n9.4628 log 5 9.2698 

logA'^ 1.7709 logA'd 1.7709 

log tan *p 9.9084 log tan d 7.0658 

a = — 13v9 log a nl.l421 6 = + O-.O log h 8.1065 

Hence we have 

Chro. middle time = 2» 59"» 55*. 

Corr. for change of lat. = — 26 .8 
Equation of eq. alts. = — 13 .9 

Chro. time app. noon = 2 59 14 .3 

At sea, instead of using the observation to find the chrono- 
meter correction, we use it to determine the ship's longitude (as 
will be fully shown hereafter) ; and therefore, to carrj' the opera- 
tion out to the end, we shall have 

Chro. time app. noon = 2* 59"* 14* 

Corr. of ehronom. = — 2 12 

Green, mean time noon = 2 57 2 

Equation of time = — 7 48 

Greenwich app. time at the local noon = 2 49 14 

which is the longitude of the middle meridian, or the longitude 
of the ship at noon. 

159. In low latitudes (as within the tropics) observations for 
the time may be taken when the sun is very near the meridian, 
for the condition that the sun should be near the prime vertical 
may then be satisfied within a few minutes of noon ; and in ease 
the ship's latitude is exactly equal to the declination, it will be 
satisfied only when the sun is on the meridian in the zenith. In 
such cases the two ecjual altitudes may be obser\'ed within a few 
minutes of each other, and all corrections, whether for change 
of latitude or change of declination, may be disregarded. 




160. By the definition, Art. 7, the latitude of a place on the 
surface of the earth is the declination of the zenith. It was also 
shown in Art. 8 to be equal to the altitude of the north pole above 
tlie horizon of the place. In adopting the latter definition, it is 
to be remembered that a depression below the horizon is a 
negative altitude, and that south latitude is negative. The 
south latitude of a place, considered numerically, or without 
regard to its algebraic sign, is equal to the elevation of the 
south pole. 

It is to be remembered, also, that the latitude thus defined is 
not an angle at the centre of the earth measured by an arc of 
the meridian, as it would be if the earth were a sphere ; but it 
is the angle which the vertical line at the place makes with the 
plane of the equator. Art. 81. 

We have seen. Art. 86, that there are abnormal deviations of 
the plumb line, which make it necessary to distinguish between 
the geodetic and the astronomical latitude. We shall here treat ex- 
clusively of the methods of determining the astronomical lati- 
tude; for this depends only upon the actual position of the 
plumb line, and is merely the declination of that point of the 
heavens towards which the plumb line is directed. 


161. Let the altitude or zenith distance of a star of known 
declination be observed at the instant when it is on the meridian. 
Deduce the true geocentric zenith distance ^, and let d be the 
geocentric declination, (p the astronomical latitude. 

Let the celestial sphere be projected on the plane of the 
meridian, and let ZNZ\ Fig. 24, be the celestial meridian: 
the centre of the sphere coincident with that of the earth ; PCP' 
the axis of the sphere; P the north pole; and ECQ the projection 


of the plane of the equinoctial. Let CZ be parallel to the 
vertical line of the obsen-er; then the point ^of the celestial 

sphere, being the vanishing point of all 
lines parallel to CZ^ is the astronomical 
zenith of the obserx'er, and ZE=^ the astro- 
nomical latitude = if. If, tfien, A is the 
position of the star on the meridian, north 
\y of the equator but south of the zenith, we 
have ZA = ^, AE = 5, and hence 

f = a + C (277) 

This equation may be treated as entirely general by attending 
to the signs of d and ^. Since in deducing it we supposed the 
star to be north of the equator, it holds for the case where it is 
south by giving the declination in that case the negative sign, 
according to the established practice; and, since we supposed 
the star to be south of the zenith, the equation will hold for the 
case where it is north of the zenith by giving ^ in that case the 
negative sign. If the star is so far north of the zenith as to be 
below the pole, or at its lower culmination, the equation will 
still hold, pro\'ided we still understand by o the star's distance 
north of the equator, measured from E through the zenith and 
ekrattd poU\ or the arc EA\ This arc is the supplement of the 
declination ; and we may here remark that, in general, any 
formula deduced for the case of a star above the pole will 
apply to the case where it is below the pole by emi»loying the 
8upi»lenu*nt of the declination instead of the declination itself; 
that is, by reckoning the declination ovtr the jyolc. 

The ease of a star beU)w the pole is, however, usually con- 
sidered under the following simple form. Put 

P = PA' = the star's polar distance, 
/* = yA' = ** true altitude, 


^ =P-{-h (278) 

in whii'h for south latitude P must be the star's south polar dis- 
tance, and the sum of P and h is only the numerical value of f . 

The declination is to be found for the instant of the meridian 
transit bv Art. GO or 02. 

In the obser\'atory, instruments are employed which give 


directly the zenith distance, or its supplement, the nadir distance. 
"With a meridian circle perfectly adjusted in the meridian^ the 
instant of transit would be known without reference to the 
clock, and the observation would be made at the instant the 
star passed the middle thread of the reticule ; but when the in- 
strument is not exactly in the meridian, or when the observation 
is not made on the middle thread, the observed zenith distance 
must be reduced to the meridian, for which see Vol. 11., Meridian 

"With the sextant or other portable instruments the meridian 
altitude of a fixed star may be distinguished as the greatest 
altitude, and no reference to the time is necessary. But, as the 
sun, moon, and planets constantly change their declination, 
their greatest altitudes may be reached either before or after the 
meridian passage ;* and in order to observe a strictly meridian 
altitude the clock time of transit must be previously computed 
and the altitude observed at that time. 

Example 1. — On March 1, 1856, in Long. 10* 5* 32* E., suppose 
the apparent meridian altitude of the sun's upper limb, north of 
the zenith, is 63° 49' 50", Barom. 30. in., Ext. Therm. 50° ; what 
is the latitude ? 

App. zen. diat. Q_ — 

26» W 10". 

r — 

+ 28 .7 

P — 

— 8 .8 

8 = 

+ 16 10 .3 

C =■ 

- 26 26 45 .2 

d = 

— 7 33 5 .8 

^ = — 33 59 51 .0 

Example 2. — July 20, 1856, suppose that at a certain place 
the true zenith distances of a Aquilce south of the zenith, and 
a Cephei north of the zenith, have been obtained as follows : 

a Aquilm a Cephei 

C = + 26^ 34' 27".5 C = — 26^ 54' 28".3 

d = + 8 29 22 7 ^ = + 6 1 58 21 .1 

^ = + 35 3 50 .2 ^ = + 35 3 52 .8 

The mean latitude obtained by the two stars is, therefore, 
f = + 35° 3' 51".5. In this example, the stars being at nearly 

* See Art. 172 for the method of finding the time of the 8un*8 greatest altitude, 
which may also be used for the moon or a planet. 
Vol. I.— 16 

the same zenith distance, but on opposite sides of the zenith, my 
constant though unknown error oj" the iiiefrument, peculiar to 
that zenith diBfance, is eliminated in tJiking the meiiD. Thus, 
if tlie zenith distance in both caees had been 10" greater, we 
should have found from a Aqtiila: f = 35* 4' 0".2, but from 
a CepUi p = 35° 3' 42".8, but the mean would still be ^ = 35* 3' 

It is evident, also, that errors in the refraction, whether due to 
the tables or to constant errors of the barometer and tliemio- 
meter, or to anj peculiar state of the air common to the t^vo 
observations, are nearly or quite eliminated by thus combining a 
pair of stars the mean of whose declinations is nearly equal to 
the declination of the zenith. The advantages of such a com- 
bination do not end here. If we select the two stars so that the 
diilerence of their zenith distances is so small that it may be 
measured with a micrometer attached to a telescope which is eo 
mounted that it may be successively directed upon the two stars 
without disturbing the angle which it makes with the vertical 
line, we can dispense altogether with a graduated circle, or, at 
least, the result obtained will be altogether independent of its 
indicatiouB. For, let J and f ' be the zenith distances, i and i' 
the declinations of the two stars, the second of which is north of 
the zenith; then, if J' denotes only the numerical value of the 
zenith distance, we have 

the mean of which is 

f = U3 + + i(c-r') 


60 that the result depends only upon the ^ven declinations and 
the observed difference of zenith distance which is measured with 
the micrometer. Such is the simple principle of the method first 
introduced by Captain Talcott, and now extensively nsed in thin 
country. To give it full effect, the instrument formerly known 
as the Zenith Telescope in England baa received several imiiortaot 
modifications from our Coast Survey. It will he fully treated of, 
in its present improved form, in Vol. II., where also will be 
found a discussion of Talcott's method in all its details, 

162. Meridian nlliludes of a eircnrnpofar stnr observed both abort 
and btlotc the pole. — Every star whose distance from the elevated 


pole is less than the latitude may be observed at both its upper 
and lower culminations. If we put 

h = the true altitude at the upper culmination, 
Aj= " '' *' lower " 

p = the star's polar distance at the upper culmination, 
p^ = " " " " lower « 

we have, evidently, 

the mean of which is 

9 = H^ + K) + HPi-P) (280) 

whence it appears that by this method the absolute values of p 
and j?! are not required, but only their difference p^ — p. The 
change of a star's declination by precession and nutation is so 
small in 12* as usually to be neglected, but for extreme precision 
ought to be allowed for. This method, then, is free from any 
error in the declination of the star, and is, therefore, employed 
in all fixed observatories. 

Example. — ^With the meridian circle of the !N*aval Academy 
the upper i^nd lower transits of Polaris were observed in 1853 
Sept. 15 and 16, and the altitudes deduced were as below: 

Upper Transit. Lower Transit. 

Sept. 16, App. alt. 40O28'26".42 Sept. 16, ST® 81' 89". 76 

Barom. 80.005 ^ Barom. 80. 146 ^ 

Att. Therm. 66<>.2 I Ref. 16 .84 Att. Therm. 75«» [ Ref. 1 12 .46 

Ext. " 68 .8 3 Ext. " 74 .63 

A = 40 27 19 .08 h^ = 87 80 27 .81 

p = 1 28 26 .04 p^= 1 28 26 .86 

= 88 68 68 .04 ^ = 88 58 63 .17 

*' " 63 .04 

Mean ^ = 88 68 68 .11 

In order to compare the results, each observation is carried 
out separately. By (280) we should have 

i{h + A,) = 38^ 58' 53".20 

i(j>i'-P)= — 09 

f = 38 58 53 .11 

This method is still subject to the whole error in the refhiction, 


which, however, in the present state of the tables, will usually be 
rery small. 

If the latitude is greater than 45°, and the star's declination 
less than 45°, the upper transit occurs on the opposite side of the 
zenith from the pole. In that case h must still represent tlie 
distance of the star from the point of the horizon below the pole, 
and will exceed 90°. Thus, among the Greenwich observations 
we find 

1837 Juno 14, Capella h^ = 7° 18' 7".94 

h = 95 39 7 .91 

q> =bl 28 37 .93 

163. Meridian zenith distances of the snn observed near the summer 
and winter solstices. — ^When the place of observation is near the 
equator, the lower culminations of stars can no longer be ob- 
served, and, consequently, the method of the preceding article 
cannot be used. The latitude found from stars observ^ed at their 
upper culminations only is dependent upon the tabular declina- 
tion, and is, therefore, subject to the error of this declination. If, 
therefore, an obsen^atory is established on or near the equator, 
and its latitude is to be fixed independently of observations made 
at other places, the meridian zenith distances of stars cannot be 
employed. The only independent method is then by meridian 
observ'ations of the sun near the solstices. 

Let us at first suppose that the obser\'ations can be obtained 
exactly at the solstice, and the obliquity (e) of the ecliptic is 
constant. The declination of the sun at the summer solstice is 
= -i- £, and at the winter solstice it is --^ — e ; hence, from the 
meridian zenith distances ^ and ^' observ^ed at these times, we 
should have 

y = :' — € 

the moan of which is 

sp = J (: + :') 

a result dependent only upon the data furnished by the obsen'a- 

Xow, the sun will not, in general, pass the meridian of the 
observer at the instant of the solstice, or when the declination is 
at its maximum value e; nor is the obliquity of the ecliptic con- 
stant. But the changes of the declination near the solstices are 
very small, and hence are verj- accurately obtained from the 


solar tables (or from the Ephemeris which is based on these 
tables), notwithstanding small errors in the absolute value of the 
obliquity. The small change in the obliquity between two 
solstices is also verj' accurately known. If then Ae is the un- 
known correction of the tabular obliquity, and the tabular values 
at the two solstices are e and e', the true values are e + Ae and 
e' + AS ; and if the tabular declinations at two observations near 
the solstices are e — x and — (e' — x'), the true declinations will 
be 5 = e + Ae — X and 5' = — (e' + Ae — x'), and by the formula 
f> = ^ + 5 we shall have for the two observations 

^ = C -{- S -\- ^€ X 

9» = C' — e' — Ae + a:' 

the mean of which is 

^ = i (c + o + J (^ - O - K^ - ^) 

a result which depends upon the small changes e — e' and x — x% 
both of which are accurately known. 

It is plain that, instead of computing these changes directly, it 
suffices to deduce the latitude from a number of observations 
near each solstice by employing the apparent declinations of the 
solar tables or the Ephemeris ; then, if <p' is the mean value of 
the latitude found from all the observations at the northern 
solstice, and f" the mean from all at the southern solstice, the 
true latitude will be 

Every observation should be the mean of the observed zenith 
distances of both the upper and the lower limb of the sun, in 
order to be independent of the tabular semidiameter and to 
eliminate errors of observation as far as possible. 


164. At the instant when the altitude is observed, the time is 
noted by the clock. The clock correction being known, we find 
the true local time, and hence the star's hour angle, by the 

t = e —a 

in which © is the sidereal time and a the star's right ascension. 


K the Bun is observed, i is simply the apparent solar time. We 
have, then, by the first equation of (14), 

sin f sin ^ -f cos f cos S cos t = sin A 

in which f is the only unknown quantity. To determine it, 
assume d and J) to satisfy the conditions 

d sin B = sin S 

d cos D = cos d cos t 

then the above equation becomes 

d cos (f — ^) = sin h 

which determines <p — D^ and hence also tp. For practical con- 
venience, however, put 

then, by eliminating d, the solution may be put under the follow- 
ing form : 

tan D = tan d sec t 

cos 7' =r sin A sin 2> cosoc d ) (281) 

<p = D dtr 

The first of these equations fully determines D, which will be 
taken numerically less than 90°, positive or negative according 
to the sign of its tangent. As t should always be less than 90°, 
or Q^y D will have the same sign as d. 

The second equation is indeterminate as to the sign of j^ 
since the cosine of + ;' and — y are the same. Hence we 
obtain by the third equation two values of the latitude. Only 
one of these values, however, is admissible when the other is 
numerically greater than 90°, which is the maximum limit of 
latitudes. When both values are within the limits + 90° and 
— 90°, the true solution is to be distinguished as that which 
agrees best with the approximate latitude, which is always suffi- 
ciently well known for this purpose, except in some peculiar 
eases at sea. 

Example 1. — 1856 March 27, in the assumed latitude 23° S. 
and longitude 43° 14' W., the double altitude of the 8un*8 lower 


limb observed with the sextant and artificial horizon was 114*^ 
40' 30'' at 4* 21"* 15* by a Greenwich Chronometer, which wa^ 
fast 2r 30*. Index Correction of Sextant = — 1' 12", Barom. 
29.72 inches, Att Therm. 61^ F., Ext. Therm. 61° F. Required 
the true latitude. 

Sextant reading = 114® 40' 30" Chronometer 4* 21"» 15- 

Index eorr. = — 1 12 Correction — 2 30 

114 39 18 

Gr. date, March 27, 4 18 45 

App. alt. Q — 57 19 39 

Longitude — 2 52 56 

Semidiameter — + 16 3 

Local mean t. — 1 25 49 

Eef. and par. = — 31 

Eq. of time — 5 19 

A— 57 35 11 

App. time, t = 1 20 30 

a — +2 51 30 

— 20° 7' 30" 

log sec t 0.027360 

log tan d 8.698351 

log tan D 8.725711 log cosec <J 1.302190 

log sin D 8.725098 
• D=+ 3® 2' 38" log sin A 9.926445 

r = 25 58 49 log cos r 9.953733 

D — 7- = SP = — 22 56 11 

Example 2. — 1856 Aug. 22; suppose the true altitude of 
Fomalhmit is found to be 29° 10' 0" when the local sidereal time 
is 21* 49-* 44- ; what is the latitude ? 

We have a = 22* 49- 44-, whence f= — l*0-0';a= — 30^22' 47".5; 
i)= — 31° 15' 13", /- = db 60° 0' 6", 9) = + 28° 44' 53". The nega- 
tive value of Y hero gives <p = — 91° 15' 19"; which is inadmissible. 

165. The observation of equal altitudes east and west of the 
meridian may be used not only for determining the time (Art. 
139), but also the latitude. For the half elapsed sidereal time 
between two such altitudes of a fixed star is at once the hour 
angle required in the method of the preceding article. When 
the sun is used in this way, the half difterence between the 
apparent times of the observations is the hour angle, and the 
declination must be taken for noon, or more strictly for the 
mean of the times of observation. By thus employing the 
mean of the A.M. and P.M. hour angles and the mean of the 
corresponding declinations, we obtain sensibly the same result 


as by computing each observation separately with its proper 
hour angle and declination and then taking the mean of the 
two resulting latitudes; and an error in the clock correction 
does not aftect the final result. The clock rate, however, must 
be known, as it affects the elapsed interval. See also Art 182. 

166. Effect of errors in the data upon the latitude computed from an 
observed altitude. — From the first of the equations (51) we find 

, dZ sinfl'cos^ ,^ , cos or ,^ 

dc> = ? — - — at A 5- dd 

cos A cos^l cos -A 

or, since h = 90° — ^^ dh = — rf^, and sin q cos d = cos f sin -4, 

d<p z= — sec A,dh — cos ^ tan A,dt -{■ cos q sec A . dd 

whence it appears that errors of altitude and time will have the 
least cftect when -4 = or 180°, that is, when the observation is 
in the meridian, and the greatest eftcct when the observation is 
on the prime vertical. K the same star is observed on both 
sides of the meridian and at equal distances from it, the coeffi- 
cient oi dt will have opposite signs at the two ob8er\'ation8, and 
hence a small error in the time will be wholly eliminated by 
taking the mean of the values of the latitude found from two 
such observations. It is advisable, therefore, in taking a series 
of observations, to distribute them symmetrically with respect to 
the meridian. Wlien they are all taken verj- near to the meri- 
dian, a special method of reduction is used, which will be 
treated of below as our Tlnrd Method of finding the latitude. 

The sign of sec A is difterent for stars north and south of 
the zenith: hence errors of altitude will be at least partially 
eliminated bv takinuj the mean of the results found from stars 
near the meridian, both north and south of the zenith. A 
constant error of the instrument may thus be \rholhj eliminated. 

As for the effect of the error do^ its coeflicient is zero onlv 

when q -- 90° and sec A is not infinite. Tliis occurs when a 

circumpolar star is obscned at its elongation, where we liave. 

Art. 18, 

, cos & 

QQQ yl — _: 

|. [sin {r, -f- y-) sin (<J — y-)] 

which shows that sec A diminishes as d increases. In order, 
therefore, to re<luce the cftect of an error in the declination 


at the same time with that of errors of altitude and time, we 
should select a star as near the pole as possible, and observe it 
at or near its greatest elongation, on either side of the meridian. 
The proximity of the star to the pole enables us to facilitate the 
reduction of a series of observations, and we shall therefore 
treat specially of this case as our Fourth Method below. 

167. When several altitudes not very far from the meridian are 
observed in succession, if we \vish to use their mean as a single 
altitude, the correction for second differences (Art. 151) must be 
applied. It is, however, preferable to incur the labor of a sepa- 
rate reduction of each altitude, as we shall then be able to com- 
pare the several results, and to discuss the probable errors of the 
observations and of the final mean. Wlien the observations are 
very near to the meridian, this separate reduction is readily 
effected, with but little additional labor, by the following method : 



168. To reduce an altitude^ observed at a given time^ to the meridian. — 
This is done in various ways. 

(A.) If in the formula, employed in Art. 164, 

sin <p sin d -\- cos <p cos 5 cos i = sin A 

we substitute 

cos ^ = 1 — 2 sin' i t 
it becomes 

sin ^ sin ^ -f cos <p cos d — 2 cos f cos d sin' J ^ = sin A 


sin ^ sin 5 -{- cos <p cos d = cos (^ — S) or cos (d — ip) 

Hence, if we put 

Cj = sp — 5, or d = ^ — s^ 

the above equation may be written 

cos C, = sin A + cos q> cos d (2 sin' i f) (282) 

If the star does not change its declination, ^j is the zenith 
distance of the star at its meridian passage ; and, being found by 


this equation, we then have the latitude as from a meridian 
observation by the formula 

SP = ^ + Ci, or ^ = a — Ci 

according as the zenith is north or south of the star. 

Wlien the star changes its declination, this method still holds 
if we take d for (he time of observaiion, as is evident from our 
formulae, in which S is the declination at the instant when the 
true altitude is h. 

To compute the second member, a previous knowledge of the 
latitude is necessaiy. As the term cos f cos 8 (2 sin* J t) de- 
creases with /, if the observations are not too far from the 
meridian, the error produced by using an approximate value of 
y will be relatively small, so that the latitude found will be a 
closer approximation than the assumed one ; and if the computa- 
tion be repeated wdth the new value, a still closer approximation 
may be made, and so on until the exact value is found. 

This method is only convenient where the computer is pro- 
vided with a table of natural sines and cosines, as well as a table 
of log. versed sines, or the logarithmic values of 2 sin' J L 

Example. — Same as Example 1, Art. 164. h = 57® 35' 11", 
J = + 2° 51' 30", i = 1* 20"' 30*. Approximate value of y = -- 23**. 

log (2 sin» J t) 8.785726 

log cos <p 9.964026 

nat. cos h 0.844201 log cos d 9.999459 

nat. no. 0.050182 log 8.749211 

nat. cos :, 0.000:^^3 

:, = — 25° 47' 54" (zenith south of sun.) 

d --^ ^ 2 51 30 

^ = — 22 "50^4 

diflering but 13" from the true value, although the assumed 
latitude was in eiTor nearly 4'. Kepeating the computation with 
— 22° 56' 24" as the approximate latitude, we find f = — 22° 56' 11", 
exaetlv as in Art. 164. 

169. (B.) AVe may also compute directly the reduction of the 
objiened altitude to the meridian altitude. Putting 

A, z^ meridian altitude = 90° — C, 



the formula (282) gives 

sin h^ — sin A = 2 cos f cos <J sin" } t 
But we have 

sin Aj — sin A = 2 cos J (h^ + h) sin i (h^ — h) 
and hence 

sin } Ch. — A) = (283) 

^ * "^ cos J (^1 + A) 

which gives the diflTerence Aj — A, or the correction of A to reduce 
it to Ai ; but it requires in the second member an approximate 
value both of tp and of A^, the latter being obtained from the 
assumed value of if by the equation A^ = 90° — (^ — S)\ or, if 
the zenith is south of the star, by the equation A^ = 90° — (5 — ^). 

Example. — Same as the above. 

a = 2^ 61' 30" logsin«i< 8.484696 

Approx. f = — 23 00 00 log cos tp 9.964026 

Cj = 26 61 30 log cos a 9.999459 

\= 64 8 30 log sec i (A, + A) 0.312573 

a J (^^ + A) = 60 51 50 log sin i (A, — A) 8.760754 

Aj — A = 6 36 33 

A = 57 35 11 a = 2° 51' 30" 

\= 64 1144 C^ = ~ 25 48 16 

f = — 22 66 46 

This method does not approximate so rapidly as the preceding, 
but the objection is of little weight when the observations are 
very near the meridian. On the other hand, it has the great 
advantage of not requiring the use of the table of natural sines. 

170. (C.) CircumTneridian altitudes. — ^When a number of altitudes 
are observed very near the meridian,* they are called circum- 
meridian altitudes. Each altitude reduced to the meridian gives 
nearly as accurate a result as if the observation were taken on the 

An approximate method of reducing such observations with 
the greatest ease is found by regarding the small arc \^{h^ — A) 
as sensibly equal to its sine ; that is, by putting 

sin } (A, — A) = i (A, — A) sin 1" 
* How near to the meridian wiU be determined in Art. 175. 


and taking h^ for } {h^ + A), from which it difters very little, 80 
that (283) may be put under the form 

, . co8c>co8^ 2 8in*}f /-ftojN 

^1 — ^ = 1 T-^rjT (284) 

cos Aj sm 1" 

The value in seconds of 

2 sin* i t 

VI = 

sin 1" 

is given in Table V. with the argument t. If h\ h'\ A'", &c. are 
the observed altitudes (corrected for refraction, etc.); <',<",/"', 
&c., the hour angles deduced from the observed clock times; 
m', m", m'", &c., the values of m from the table ; and we put the 
constant factor 


. cos <p cos d cos ip cos d 

cos A, sin Cx 

we have A, = A' + Am' 

A, = A" + Aw!' 
\ = A'" + Am'" 

and the mean of all these equations gives 

h' J- h" + A'" + etc. , . m' + m" + m'" + &c. 
A = 1- A 

n n 

in which n is the number of observations ; or 

h, = h^ + Am, (286) 

in which A^, denotes the mean of the observed altitudes corrected 
for refraction, &c., and m^ the mean of the values of m. 

When Aj has been thus found, the latitude is deduced as from 
any meridian altitude, only ob8er\'ing that for the sun the de- 
clination to be used is that which corresponds to the mean of 
the times of observation, as has already been remarked in Art. 

Example. — At the U. 8. Naval Academy, 1849 June 22, cir- 
cummeridian altitudes of ^9 Ursac 3Iinoris were observed with a 
Troughton sextant from an artificial horizon, as in the following 
table. The times were noted by a sidereal chronometer which 


was fast 1* 45'.7. The index correction of the sextant was 
— 14' 58", Barometer, 30.81 inches, Att. Therm. 65° F., Ext. 
Therm. 64° F. 

The right ascension of the star was 14* 51* 14*.0 
Chronometer fast +1 45 .7 

Chronometer time of star's transit 14 52 59 .7 

The hour angles in the column t are found by taking the differ- 
ence between each observed chronometer time and this chro- 
nometer time of transit. 

2 Alt. :i|c 




108O 89* 40" 

14* 45^ 47». 

7* 12».7 


89 50 



5 58.7 


40 40 



4 5.2 





1 80.2 





1 86.8 


40 80 



8 22.8 


40 20 



4 48.8 





5 47.8 





7 17.8 


89 20 



9 10.8 



108 40 14 

jii0 = 

= 61.68 

Ind. corr. 

— 14 58 

108 25 16 

Assumed ^ — 



54 12 88 

6 — 


46 86 .9 


— 42.0 

Approx. Ci = 


47 86 .9 


COB ^ 9.8906 


+ 21 .5 


008 6 9.4198 


54 12 17 .6 


cosecCi 0.2829 


- 86 47 42 .6 


A 9.5428 

6 — 

74 46 86 .9 


m^ 1.7898 

^ - 

88 58 54 .4 


Am^ 1.8826 

Remark 1. — The reduction h^ — h increases as the denominator 
of A decreases, that is, as the meridian zenith distance decreases. 
The preceding method, therefore, as it supposes the reduction to 
be small, should not be employed when the star passes very near 
the zenith, unless at the same time the observations are restricted 
to very small hour angles. It can be shown, however, from the 
more complete formulsB to be given presently, that so long as 
the zenith distance is not less than 10°, the reduction computed 
by this method may amount to 4' 30" without being in error 
more than 1" ; and this degree of accuracy suffices for even the 
best observations made with the sextant. 


Remark 2.— K in (284) we put sin J < = V sin V. t {t being in 
seconds of time), we have 

, - cos f cos ^ 225 . - „ ., ., .^^^ 

Ai — A = sm 1". ^' = at* (287) 

cos h^ 2 

in which a denotes the product of all the constant factors. It 
follows from this formula that near the meridian the altitude varies 
as the square of the hour angle, and not simply in proportion to the 
time. Hence it is that near the meridian we cannot reduce a 
number of altitudes by taking their mean to correspond to the 
mean of the times, as is done (in most cases without sensible 
error) when the observations are remote from the meridian. 
The method of reduction above exemplified amounts to sepa- 
rately reducing each altitude and then taking the mean of all 
the results. 

171. (D.) Circummeridian altitudes more accurately reduced. — The 
small correction which the preceding method requires will be 
obtained by developing into series the rigorous equation (282). 
This equation, when we put ^ = 90° — h = true zenith distance 
deduced from the observation, may be put under the form 

cos C = cos Ci — 2 cos ^ cos d sin* J t 

which developed in series* gives, neglecting sixth and higher 
powers of sin J /, 

* If we put y = 2 cos ^ cos 6 sin' } t, the equation to be dcTeloped is 

cos c = cos Ci — y (a) 

in which ^^ is constant and ^ may be regarded as a fiinotion of y ; so thai bj Mac- 
LAUBiN^s Theorem 

in which (/), | — j. &c. denote the Talues of /y and its differential coefficient* when 
y ^=r 0. The equation (a) gires, by differentiation, 

sin ^ — =1 — = 

</y rfy sin ^ 

rf*C COS C <^ cot c 

rfy' sin' ^ dy sin» ^ 





ip COS ^ 2 sin* \ t I cos y cos d \« 2 cot Ci sin* j t . g 
;hr^^ sinl" I sinCi / sin 1" 

By this formula, first given by DelambRb, the reduction to 
the meridian consists of two terms, the first of which is the same 
as that employed in the preceding method, and the second is the 
small correction which that method requires. These two terms 
will be designated as the " Ist Eeduction" and " 2d Reduction." 

we have 

2 sin« i t 
m — 

sin 1" 

2 sin' } t 

n — 

sin 1" 

- cos cos d 

A — 

B — A* cot Ci 

sin Ci 

Ci c 

— Am 




If a number of observations are taken, we have a number of 
equations of this form, the mean of which will be 

Ci = Co — Am^ + Bn^ 

in which ^j, is the arithmetical mean of the observed zenith dis- 
tances, niQ and n^ the arithmetical means of the values of 7n and 
71 corresponding to the values of t. The values of n are also 
given in Table V. 

Having found (^j, we have the latitude, as before, by the formula 

^ = ^ + C, 

in which we must give (^^ the negative sign when the zenith is 
south of the star, and it must be remembered that for the sun 
(or any object whose proper motion is sensible) 8 must be the 
mean of the declinations belonging to the several observations. 

But when y = we have, by (a), ^ = Ci> bo that (6) becomes 

y y' cot ^, «• 

BeBtoring the Talue of y, this giYes the deyelopment used in the text, obseiring that 
as Z and ^^ are supposed to be in seconds of arc, the terms of the series are divided 
by sin 1'' to reduce them to the same unit. 


or, which is the same, the declination corresponding to the mean 
of the times of observation.* 

Finally, if the star is near the meridian below the pole, the 
hour angles should be reckoned from the instant of the lower 
transit. Eecurring to the formula 

cos C = sin ^ sin d -\- cos f cos d cos t 

in which t is the hour angle reckoned from the upper transit, 

we observe that if this angle is reckoned from the lower transit 

we must put 180° — i instead of /, or — cos i for + cos U and theu 

we have 

cos C = sin ^ sin d — cos f cos d cost 

and, substituting as before, 

cos i = 1 — 2 sin" } t 
this gives 

cos C = — cos (f + ^) + 2 cos f cos d sin" i t 

or, since for lower culminations we have f^ = 180*^ — (s^ + ^ 
and cos ^^ = — cos (^ + 5), 

cos C = cos C, + 2 cos f cos d sin* J t 

which developed gives 

^ , cos ^ cos d 2 sin* it / cos f cos d^ 2 cot Ti sin* } t 

* * sin Ci sin 1" \ sin Ci / sin 1" 


Ci = C + ilm + J?n (sub polo) (290) 

which is computed by the same table, but both first and second 
reductions here have the same sign. 

If a 8tar is obsen-ed with a sidereal chronometer the daily 
rate of which is so small as to be insensible during the time of 

* To Bhow that the mean declination is to be used, we maj obserre that for each 
observation we have put Ci = ^ — (5, and that if d\ d'\ &c., are the seTenJ deelin*- 
tions, the seTeral equations of the form (289) will give 

^ = <J' 4- ;' — ^m' -f A* cot ^, n' 
^ = <J" + ^' — Am'f -f A^ cot ^j n" 

the mean of which, if <J = mean of S\ 6'\ &c., will be 

^ = <J + (• — -4wo H- -i* cot Ci fi, == <J -f. fj 


the observations, the hour angles i are found by merely taking 
the difference between each noted time and the chronometer 
time of the star's transit, as in the example of Article 170. But 
if we wish to take account of the rate of the chronometer, it can 
be done without separately correcting each hour angle, as fol 
lows: Let 8 The the rate of the chronometer in 24* (57 being 
positive for losing rate. Art 137) ; then, if < is the hour angle 
given directly by the chronometer, and i' the true hour angle, 
we have 

f:t = 24»: 24»— dT= 86400' : 86400- — dT 


f = t 

L S6400j 

Instead of sin J< we must use sin J<'; for which we shall have, 
with all requisite precision, 

sin } ^ = sin } ^ . — , or sin* if = sin' } ^ . j - j 
Hence, if we put 

we shall have 

. , cos cp cos ^ 2 sin' } t 

sm Ci sm 1" 

so that if we compute A by the formula 

cos f cos d 


sin Ci 

we can take m = — ; — :rr- for the actual chronometer intervals, 

sm 1" ' 

and no further attention to the rate is required. 

The factor k can be given in a small table with the argument 
"rate," in connection with the table for m, as in our Table V. 

K a star is observed with a mean time chronometer, the inter- 
vals are not only to be corrected for rate, but also to be reduced 

Vol. L— 1« 


from mean to sidereal intervals by multiplying them by ;£ = 
1.00273791 (Art. 49) ; so that for sin* i t we must substitute k sin* 
(J./i^), or, with sufficient precision, kff sin* } L 

K the sun is observed with a mean time chronometer, the in- 
tervals are both to be corrected for rate and reduced from mean 
solar to apparent solar intervals. The mean interval differs 
from the apparent only by the change in the equation of time 
during the interval, and this change may be combined with the 
rate of the chronometer. Denoting by 8E the increase of the 
equation of time in 24* (remembering that -E is to be regarded 
as positive when it is additive to apparent time), and by ^Tthe 
rate of the chronometer on mean time, we may regard dT — dE 
as the rate of the chronometer on apparent time. Instead of 
the factor k we shall then have a factor A;', which is to be found 
by the formula 

L "~ 86400 

which may be taken from the table for k by taking dT — dE as 
the argument. 

Finally, if the sun is observed with a sidereal chronometer, 

we must multiply sin* J t not only by A' but by -\, 

Denoting // by i and — , by t', these rules may be collected, for 
the convenience of reference, as follows: 

Star by sidereal chron., ^ = A* . -, 

fiin Cj 

Star b\' mean time cbron.,^ = Ai — ' .^^ [log t= 0.002375] 

sin C, 


r, 1 .1 J f . COS <r cos r5 

Sun by mean time chron., A =: k ; 

sin Ci 

Sun by sidereal chron., A=^h! i' ^^'1?L5^1_ [log a = 9.997625] 

sin «»| 

for which log k will be taken from Table V. with the argument 
rate of the chronometer = dT\ and log k' from the same table 


with the argnment dT — dE=^ daily rate of the chronometer 
diminished by the daily increase of the equation of time. 

Example. — 1856 March 15, at a place assumed to be in lati- 
tude 87° 49' N. and longitude 122° 24' W., suppose the fol- 
lowing zenith distances of the sun's lower limb to have been 
observed with an Ertel universal instrument,* Barom. 29.85 
inches, Att Therm. 65° F., Ext. Therm. 63° F. The chrono- 
meter, regulated to the local mean time, was, at noon, slow 
11- 20*.8, with a daily losing rate of 6'.6. 

m n 

Obs'd xeD. dist. Chronometer. t 

40° 8'40".7 23»37-85*. — 19«58'.8 788".8 1".49 

40 2 16.5 42 3. —15 80.8 472.4 0.54 

39 57 28 .8 46 29.5 —11 4.8 240 .6 .14 
89 54 17 .2 50 46.5 — 6 47.8 90 .5 .02 
89 52 88 . 55 16. — 2 17.8 10 .4 .00 
89 52 84 .5 87 .5 + 8 8 .7 18 .4 .00 
89 54 28 .6 5 18 . 7 89 .2 115 .0 .08 
89 58 9 .8 9 49.5 12 15.7 295.1 0.21 

40 8 .8 14 8 . 16 34 .2 588 .9 .70 
40 9 86 . 18 81 . 20 57 .2 861 .4 1 .80 

Means 89 59 18 .5 U= +0 29.1 7?io=842 .60no=0 .49 

The equation of time at the local noon being + 8* 54*.6, we 


Mean time of app. noon = 0* 8* 54*.6 
Chronometer slow = 11 20.8 

Chr. time of app. noon = 28 57 88 .8 

The difference between this and the observed chronometer 
times gives the hour angles t as above. 

The mean of the hour angles being + 29*.l, the declination is 
to be taken for the local apparent time 0* 0* 29^.1, or for the 
Greenwich mean time March 15, 8* 18"* 59^.7; whence 

^ = — 1<> 48' 8".8 
(Approximate) ^ = -f- 87 49 . 
" Ci= 89 87 8 .8 

The increase of the equation of time in 24* is dE = — 17'.4, 

* See Vol. IL, Altitude and Azimuth Instrument^ for the method of obserTing the 
senith dieianoes. 


and, the chronometer rate being 8T = + 6\6, we have dT — 8E 
=^ + 24*. 0, with which as the argument "rate" in Table V. we 
find log A' = 0.00024. 

The computation of the latitude is now carried out as follows: 

log cos f 9.89761 Mean observed zen. dist. Q = 39® 59^ 18''.5 

log cos d 9.99979 r—p= + 41 .8 

log cosec :, 0.19540 log A* 0.1861 S= — 16 6 .5 

log ^ 0.00024 log cot C, 0.0821 ^iWp = — 7 4 .4 

log^ 0.09304 logJB 0.2682 Bno= + .9 

log wio 2.53479 log no 9.6902 C, = 39 36 60 .3 

logilmo 2.62783 log J?no 9.9584 d= — 1 48 8 .8 

f = 37 48 41 .5 

The assumed value of y being in error, the value of A is not 
quite correct; but a repetition of the computation with the value 
of <p just found does not in this case change the result so much 
as O'M. 

172. (E.) Gauss's method of reducing circummeridian altitudes of 
the sun. — The preceding method of reduction is both brief and 
accurate, and the latitude found is the mean of all the values 
that would be found by reducing each ob8er^'ation separately. 
This separate reduction, however, is often preferred, notwith- 
standing the increased labor, as it enables us to compare the 
observ'ations with each other, and to discuss the probable error 
of the final result; and it is also a check against any gross error. 
But, if we separately reduce the observations by the preceding 
method, we must not only inteqiolate the refraction for each 
altitude, but also the declination for each hour angle. Gavss 
proposed a method by which the latter of these inteqmlations is 
avoided. He showed that if we reckon the hour angles, not 
from apparent noon, but from the instant when the sun reaches its 
warinmm altitude^ we can compute the reduction by the method 
above iriven, and use the meridian declination for all the obserx'a- 
tions. This method is, indeed, not quite so exact as the preced- 
ing; but I shall show how it nniy be rendered (piite perfect in 
practice by the introduction of a small correction. 

In the rigorous formula 

cos C = sin <f sin H + cos f cos d cos t 


d is the declination corresponding to the hour angle t. If then 

aJ = the hourly increase of the declination, ^positive when 

the sun is moving northward^ 
d^ = the declination at noon, 

and if ^ is expressed in seconds of time, we have 

t . Ad 
* ^ 8600 ' ^ 

where, since ^d never exceeds 60", x will not exceed 80" so long 
as < < 80*. Hence we may suhstitute, with great accuracy, 

sin d = sin d^ -|- cos d^ sin x 

cos J = cos dj — sin d^ sin a: 

and the ahove formula becomes 

cos C = sin 5? sin d^ + cos f cos ^^ cos f + sin (^ — d^) sin x 

4- 2 COB f sin dj sin' } ^ sin or 

The last term is extremely small, rarely affecting the value of ^ 
by as much as 0".l ; and since x is proportional to the hour 
angle, and therefore has opposite signs for observations on differ- 
ent sides of the meridian, the effect of this term will nearly or 
quite disappear from the mean of a series of observations pro- 
perly distributed before and after the meridian passage. Now, 
we have 

fAdsinl" i;:. . i« Ad 

sm x = — --— — = 15 ^ sm 1" . 


then, taking 

8600 54000 

. « Ad Bin(cp — O 
sm ^ = — ^ ^ 

54000 cos f> cos d^ 

15 t sm 1" = sin f + J sin» t 

we have 

smx =(sm t + i sin' t) sin i^ • — 

sin (^ — dj 

and the formula for cos ^ becomes, by omitting the last term, 

cos C = sin ^ sin d^ -\- cos f cos dj(co8 1 + sin t sin t>) 

-(- } cos ^ cos d^ sin'^ sin t9 


The last temi involving sin' t multiplied by the small quantity 
sin I? is even less than the term above rejected. Like that, also, 
it has opposite signs for observations on different sides of the 
meridian, and will not aftect the mean result of a properly 
arranged series of observations. Rejecting it, therefore, our for- 
mula becomes 

cos C = sin f sin d^ -f cos ^ cos d^ cos (t — d) 

-f 2 cos ^ cos d^ sin* J d^ 

The last term here must also be rejected if we wish to obtain the 
method as proposed by Gauss ; but, as it is always a positive 
term and affects all the observations alike, I shall retain it, espe- 
cially as it can be taken into account in an extremely simple 

The maximum value of cos ^, which corresponds to the 
maximum altitude, is given immediately by the above formula 
by putting t = t?. Hence t? is the hour angle of the maximum altitude. 

we have 


cos C = cos (f — d^) — 2 cos f> cos d^ sin* } f 

+ 2 cos ^ cos d^ sin* J * 

, cos f cos d 2 sin* 1 ^ 

0=0^ -j *-. 

sin (^ — d) sin 1" 

then our formula becomes 

cos C = cos (f — 5') — 2 cos f> cos d^ sin* J f 

This equation is of the same form as that from which (288) was 
obtained, and therefore when developed gives 

cos ^ cos 5, 2 sin* If / cos tp cos ^, \* 2 cot Cj sin* } f 

* sin Cj sin 1" \ sin Cj / sin 1" 

in which Ci = ^ — ^'- Putting then, as before, 

. cos cp cos d. _ ^. . - ,/*«.^x 

A = f— — 1 B = A* cot Ci (292) 

sm ^ 


and taking m and ?i from Table V., or their logarithms from 
Table VI., with the argument t\ which is the hour angle reckoned 


from the instant the sun reaches its maximum altitude, we have 

Ci = C — Am + Bn (293) 

Since (^^ differs from the latitude by the constant quantity 5', its 
value found from each observation should be the same. Taking 
its mean value, we have 

f = C, + ^' 

The angle t>, being very small, may be found with the utmost 
precision by the formula 

^ = — = [9.40694] — (294) 

810000 sin r A *• ^ A ^ 

which gives ^ in seconds of the chronometer when A has been 
computed by the formula (292). 

The most simple method of finding the corrected hour angles 
V will be to add t? to the chronometer time of apparent noon, 
and then take the difference between this corrected time and 
each observed time. 

If we put d^ = di + y, we have 

y = A.l^ (295) 

Sin 1" 

which requires only one new logarithm to be^taken, namely, the 
value of log m from Table VI. with the argument &. We then 
have, finally, 

9 = ^^+^i + y (296) 

Example. — The same as that of the preceding article. We 
have there employed the assumed latitude 37® 49' ; biit, since even 
a rough computation of two or three observations will give a 
nearer value, let us suppose we have found as a first approxima- 
tion y = 37° 48' 45". With this and the meridian declination 
*i = — 1° 48' 9".2, and log A' = 0.00024 as before, we now find, 
by (292), 

log A = 0.09810 log B = 0.2683 

We have also there found the chronometer time of apparent 


noon = 23* 57* 33\8. We now take from the Ephemeris Ai = 
+ 59".22, and hence, by (294), 

log Ad 1.7725 
ar. CO. log A 9.9069 
const, log 9.4059 

^ = + 12'.2 log * 1.0853 

Hence the chronometer time of the maximum altitude is 
23* 57" 33'.8 + 12'.2 = 23* 57" 46', which gives the hour angles 
/' at3 below : 



log Am 



— 20- 11*. 





15 43. 





11 16.5 





6 59.5 





— 2 30. 



+ 2 51.5 



7 27. 





12 3.5 





16 22. 





20 45. 





The refraction may be computed from the tables first for a mean 
zenith distance, and then with its variation in one minute (which 
will be found with sufficient accuracv from the table of mean 
refraction) its vahie for each zenith distance is readily found. 
The parallax, which is here sensibly the same {= 6".54) for all 
the observations, is subtracted from the refraction, and the results 
are given in the column r — p of the following computation. 
The numbers in the 3d and 4th columns are found from their 
logarithms above ; and the last column contains the sevenU 
values of the minimum zenith distance of the sun's lower limb, 
formed by adding together the numbers of the preceding columns. 
To the mean of these we then apply the sun's semidiameter, the 
meridian declination, and the correction y, which are all constant 
for the whole series of observations. 


Obs'd ; 






40<» 8'40".7 

+ 42".l 

— 16' 30".5 

+ 2".9 


> 52' 55".2 

40 2 16 .5 

41 .0 

10 .7 

1 .1 

58 .8 

89 57 28 .3 

41 .8 

5 9 .2 


61 .2 

89 54 17 .2 

41 .7 

1 58 .9 


60 .0 

89 52 33 . 

41 .6 

15 .2 


59 .4 

89 52 34 .5 

41 .6 

19 .9 


56 .2 

89 54 28 .6 

41 .7 

2 15 .0 


55 .4 

39 58 9 .8 

41 .8 

5 58 .7 


58 .3 

40 3 .3 

41 .9 

10 51 .4 

1 .2 

52 .0 

40 9 36 . 

42 .1 

17 26 .8 

8 .2 

54 .5 

(Lower limb) Mean C, — 



57 .10 

, 2sin»i* 
'"« sinl" 


Semidiameter =: 


6 .49 


— 1 


9 .20 



y — 





9 — 



41 .51 

This result agrees precisely with that found before. If we suppose 
all the obsei-vations to be of the same weight, we can now deter- 
mine the probable error of observation. Denoting the difference 
between each value of {^^ and the mean of all by r, and the sum 
of the squares of v by [fr], according to the notation used in the 
method of least squares, we have 



— 1".9 


+ 1.7 


+ 4.1 


+ 2 .9 


+ 2 .3 


— .9 


— 1 .7 


+ 1.2 


— 5 .1 


-2 .6 


Mean error of a single obscrva- 

tion = Ji?^l = 2".89 
A/n — 1 

Mean error of the final value of 

2^ ^ ^,^^ 


n = 10, [vv'] = 74.92 

Probable error of a single obs. = 2".89 X 0.6745 = 1".95 
« « of sp =0 .91 X 0.6745 = .61 

It must not be forgotten that the probable error 1".95 here 
represents the probable error of observation only : a constant error 
of the instrument, affecting all the obser\^ations, will form no 
part of this error; and the same is true of an error in the 


173. For the most refined determinations of the latitude, 
standard stars are to be preferred to the sun. Their deelinatious 
are somewhat more precisely known ; the instrument is in night 
observations less liable to the errors produced by changes of 
temperature during the observations ; constant instrumental 
errors and errors of refraction may bo eliminated to a great 
extent by combining north and south stars ; or errors of declina- 
tion may be avoided by employing only circumpolar stars at or 
near their upper and lower culminations. In general, errors of 
circunmieridian altitudes are eliminated under tlie same condi- 
tions as those of meridian observations, since the former are 
reduced to the meridian with the greatest precision. See the 
next following article. 

For a great number of nice determinations of the latitude by 
circummeridian altitudes of stars north and soutli of the zenith 
and of circumpolar stars, see Puissant, Nouvclle Dcsmption Gio- 
mitriquc de la France. 

174. Effect of errors of zenith distance^ declination^ and timCj upon 
the latitude found by circunmieridian altitudes. — Difterentiating (289), 
regarding A as constant, and neglecting the variations of the 
last term, which is too small to be sensibly affected by small 
errors of /, we have, since dip —- rf^^i + ddy 

d^ = d: + dd- -4^"-^ (Ibdt) 

sm 1" 

The errors rf^ and dS aftect the resulting latitude by their whole 
amount. The coefficient of dt bus opposite signs for east and 
west hour angles; and therefore, if the observations are so taken 
as to consist of a number of pairs of equal zenith distances east 
and west of the meridian, a small constant error in tlie hour 
angles, arising from an imperfect clock correction, will be elimi- 
nated in the mean. This cimdition is in jjractice nearly satisfied 
when the same number of observations are taken on each side 
of the meridian, the intenals of time between the successive 
observations being made as nearly equal as practicable. 

An error in the assumed latitu<le which atl'ei^ts A is eliminated 
by repeating the computation with the latitude found by the first 
computation. An error in the declination which would affect A 
is not to be supposed. 


175. To determine the Umiis within which the preceding methods of 
reducing circunimeridian altitudes are applicable. — First. lu the 
method of Art. 170 we employ only the " first reduction*' (= Am\ 
which is the first term of the more complete reduction expressed 
by (288). The error of neglecting the " second reduction** (= Bn) 
increases with the hour angle, and if this method is to be used it 
becomes necessary to determine the value of the hour angle at 
which this reduction would be sensible. We have 

-D .I. X ^ 2 sin* J* 

Bn = A* cot C, ^— 

' sinl" 

whence if we put 5 for Bn and 

2?' = l/TsmTManT 
we derive 

sin« it = — i/b (298) 


Since Zi = f — 5, F and A are but functions of ^ and d ; and 
therefore by this formula we can compute the values of / for 
any assigned value of b, and for a series of values of ^ and 8. 
Table Vn.A gives the values of / in minutes computed by (298) 
when 5 = 1". That is, calling /j the tabular hour angle and t 
the hour angle for any assigned limit of error 6, we have 

sin* it^ = — sin' i t = sin' i t^ y^b 


As the limits are not required with great precision, we may sub- 
stitute for the last equation the following : 

t = t//b 

If we take b = 0".l, we have yb = 0.56, or nearly J : hence the 
limiting hour angle at which the second reduction amounts to O'M is 
about } the angle given in Table VILA. 

Example. — How fur from the meridian may the observations 
in the example p. 237 be extended before the error of the 
method of reduction there employed amounts to 1"? With 
f= + 39°, d= + 75°, Table vri.A gives /j = 30^ Hence 


the method is in that example correct within 1" if the obserra- 
tions are taken within 30" of the meridian, and correct within 
O'M if they are taken within 15* of the meridian. 

Second. — ^In the more exact methods of reduction given in 
Arts. 171 and 172, we have neglected the last term of the 
development given in the note on page 239, which may be called 
a " third reduction." Denoting it by e, we have 

c = -- 1 — ■ ^ I J.' sm* i t 

3 \ sin 1" / 

whence, if we put 

we deduce 

\l + 3cot«C, 

8in« ht = — i/c (299) 


Table Vil.B gives the values of ^ computed by this formula, for 
c = 1". Denoting the tabular value of t by /^ we have 

sin* } f, = — sin* it = sin* i t^ {/c 


or, with sufficient accuracy in most cases, 

t = t^ V'C 

• / 

For c = 0".l we have i. e = 0.68, or nearly \ ; and hence the 
limiting hour angle at which the thirtl reduction (omitted in our 
most exact methods) would amount to O'M is about § the angle 
given in Table VII.B. 

Example. — How for from the meridian may the observations 
in the example p. 243 be extended before the error of the 
method of reduction there employed amounts to O'M ? With 
ip = 38^, a = — 2°, Table VII.B gives t^ -^ 39*, and § of this 
is / = 26": so that the method is in that example correct within 
1" when the obsen-ations are taken within 31>" of the meridian ; 
and it is correct within 0".l when the obser\'ations arc taken 
\\'ithin 26"* of the meridian. 

The limiting hour angle for a given limit of error diminibhea 


rapidly with the zenith distance ; and hence in general very small 
zenith distances are to be avoided. But the observations may be 
extended somewhat beyond the limits of our tables, since the 
errors which affect only the extreme observations are reduced in 
taking the mean of a series. 


176. The latitude may be deduced with accuracy from an alti- 
tude of the pole star observed at any time whatever, when this 
time is known. The computation may be performed by (281); 
but when a number of successive observations are to be reduced, 
the following methods are to be preferred. If we put 

p = the star's polar distance, 
we have, by (14), 

sin A = sin ^ cos p -j- cos ^ sin p cos t 

in which the hour angle t and the altitude h are derived from 
observation and <p is the required latitude. Now, p being small 
(at present less than 1° 30'), we may develop y in a series of 
ascending powers of py and then employ as many terms as we 
need to attain any given degree of precision. The altitude 
cannot differ from the latitude by more than 2); if, then, we put 

^= h — X 

X will be a small correction of the same order of magnitude as p. 
We have* 

sin ^ = sin (h — ar) = sin h — x cos h — ^ x* sin A + J ^ cos h + &c. 
cos f> = cos (h — x) = cos h-^- XQinh — J x* cos h — J a;* sin A + &c. 
sinp =:p — J2^ + &c. 
cosp = 1 — Ip* + &c* 

which substituted in the above formula for sin h give 

sin A = sin A — x cosh -^ p cost cosh — ^(o:* — 2 xp cos t-\-p^) sin h-\-&c. 

and from this we obtain the following general expression of the 
correction : 

♦ PL Trig. (408) and (406). 


a: = jp COS f — ^ (.r*— 2 xp COS f + ;>') tan A 

+ 24 (^— 4 x»|> COS f + 6 x*/)*— 4 jy^cos t+p*) tan A 

— &c. (a) 

For a first approximation, we take 

x = pco&t (b) 

and, substituting this in the second term of (a), we find for a 
second approximation, neglecting the third powers of p and x, 

X = p cos t — J j>* sin* t tan A (c) 

Substituting this value in the second and third terms of (a), we 
find, as a third approximation, 

X = p cos t — Ip^ sin* t tan A + i />• cos t sin* t (d) 

This value, substituted in the second, third, and fourth terms of 
(a), gives, as a fourth approximation, 

X == p cos t — i j>*sin*< tan A + jp'cos t sin't — |/)*sin*f tan'A 

4- ^1^ p4 (4 _ 9 sin. ^) gjn* < tan A (e) 

In these formulje, to obtain x in seconds when p is given in 
seconds, we must multiply the terms in /)*, pl^y and p* by sin 1", 
sin* 1", sin' 1", respectively. 

In order to determine the relative accuracv of these formulae, 
let us examine the several terms of the last, which embraces all 
the others. The value of t, which makes the last term of (c) a 
maximum, will be found by putting the difterential coefficient 
of (4 — 9 sin* /) sin* t equal to zero ; whence 

4 sin ^ cos t (2 — 9 sin* = 

which is satisfied by ^ = 0, / = 90°, or sin* t = J, the last of which 
alone makes the second difterential coefticient negative. The 
maximum value of the term is, then, ^p* sin' l"tan A, which 
for p = 1° 30' = 5400" is 0".0018 tan A. This can amount to 
0".01 only when A is nearly 80°, — that is, when the latitude is 
nearly 80°. This term, therefore, is wholly inappreciable in 
every practical case. 


The temi }2>*8in*l" sin^^tan' Ais a maximum for 8in< = l, 
in which case, for p = 5400", it is 0".0121 tan* h. This amounts 
to O'M when h = 64^, and to 1". when h = 77°. 

For the maximum of the term J jj^ sin^ 1" cos / sin* / we have, 
by putting the differential coefficient of cos t sin* t equal to zero, 

sin ^ (2 ~ 3 sin* t) = 

which gives sin* < = f , and consequently cos / = i/ J ; and hence 

the maximum value of this term is JjD^sin* 1"|/J = 0".475. 

Since the maximum values of this and the following terms do 

not occur for the same value of /, their aggregate will evidently 

never amount to 1" in any practical case. 

Hence, (o find the latitude by the pole star tmthin 1", we have the 


^ = A — ;) cos f + i;)* sin 1" sin* t tan h (300) 

The hour angle t is to be deduced from the sidereal time 

of the observation and the star's right ascension a, by the 


^== e — a 

To facilitate the computation of the formula (300), tables are 
given in every volume of the British Nautical Almanac and the 
Berlin Jahrbuch; but the formula is so simple that a direct 
computation consumes very little more time than the use of 
these tables, and it is certainly more accurate. 

Example. — (From the Nautical Almanac for 1861). — On March 
6, 1681, in Longitude 37° W., at 7* 43"' 35* mean time, suppose 
the altitude of PolariSj when corrected for the error of the in- 
strument, refraction, and dip of the horizon, to be 46° 17' 28" : 
required the latitude. 

Mean time 

7* 43- 35*. 

Sid. time mean noon, March 6, 

22 56 47.9 

Reduction for 7* 43« 35* 

1 16.2 

Reduction for Long. 2* 28* 


Sidereal time 


^ — • 

6 42 3.4 

March 6, ;> — l'^ 25' 32".7 



1 7 39.0 


5 34 24.4 

83^36' 6" 


Hence, by formula (300), 

logp 3.71035 log|>« 7.4207 

log cost 9.04704 logsin*^ 9.9946 

log Ist term 2.75739 log tan h 0.0196 

log i sin I'' 4.3845 

h = 46° 17' 28" log 2d term 1.8194 

Ist term = — 9 32 .0 
2d « = + 1 6 .0 

SP = 46 9 2 .0 
By the Tables in the Almanac, 9> = 46** ^ 1" 

177. If we take all the terms of (e) except the last, which we 
have shown to be insignificant, we have the formula 

f = A — p cos ^ + ll>* sin 1" sin' t tan h 

^\f sin« 1" cos t sin« t + J;)* sin» 1" sin* t ten* A (301) 

which is exact within 0".01 for all latitudes less than 75®, and 
serves for the reduction of the most refined observations with 
first-class instruments. 

If we have taken a number of altitudes in succession, the 
separate reduction of each by this formula will be very laborious. 
To facilitate the operation, Petersen has computed verj^ con- 
venient tables, which are given in Schumacher's Hiilfstafdn^ 
edited by Warnstorff. These tables give the values of the 
following quantities : 

a=p^ cos t -\- \p^ sin* 1" cos t sin* t 

P =^iPo^ sin 1" sin* t 

X = Ip (;>* — ;>o*) ®*"* ^" ^^^ ' ®*^* ^ 
;i =1 ^p* sin* 1" sin* t tan* h 

in which p, = r 30' = 5400". Then, putting 


log A = logp — 3.7323938 
the formula (301) becomes 

^ = h — (^Aa + X) + A*?ianh + /i 


Putting then 

we have ' ^ ^ 

f, = h'^x + y 

or, when the zenith distance ^ is observed^ 

X =z Aa, -\- X 

y = ^«/9cotC + A >(303) 

The first table gives a with the argument t ; the second, /? with 
the argument /; the third, X with the arguments p and t; and 
the fourth, /£ with the arguments y and y, ^ being used for h in 
BO small a term. 

To reduce a series of altitudes or zenith distances by these 
tables, we take for A or f the mean of the true altitudes or 
zenith distances ; for a and /? the means of the tabular numbers 
corresponding to the several hour angles, with which we find 
Aa and A^^ cot f . The mean values of the very small quanti- 
ties X and /£ are sensibly the same as the values corresponding to 
the mean of the hour angles ; so that X is taken out but once for 
the arguments polar distance and mean hour angle, and /jl is 
taken with the arguments f and the approximate value of y = 
A*^ cot ^. Illustrative examples are given in connection with 
the tables. 


178. Let S and S'y Fig. 25, be any two points of 
the celestial sphere, Z the zenith of the observer, 
Pthe pole. Suppose that the altitudes of stars seen 
at S and iS', either at the same time or diff^ercnt 
times, are observed, and that the observer has the 
means of determining the angle SPS' ; also that 
the right ascensions and declinations of the stars 
are known. From these data we can find both the latitude and the 
local time. A graphic solution (upon an artificial globe) is indeed 
quite simple, and it will throw light upon the analytic solution. 
With the known polar distances of the stars and the angle SPS'^ 

Vol. L— it 

258 . LATITUDE. 

let the triangle SPS' be constructed. From S and S' as poles 
describe small circles whose radii (on the surface of the sphere) 
are the given zenith distances of /Sand S' : these small circles inter- 
sect in the zenith ^ of the observer, and, consequently, determine 
the distance PZ, or the co-latitude, and at the same time the hour 
angles ZPS and ZPS\ from either of which with the star's right 
ascension we deduce the local time. This graphic solution shows 
clearly that the problem has, in general, two solutions ; for the 
small circles described from S and S' as poles intersect in two 
points, and thus determine the zenith of another observer who 
at the same instants of time might have observed the same alti- 
tudes of the same stars. The analytic solution will, therefore, 
also give two values of the latitude; but in practice the ob- 
server always has an approximate knowledge of the latitude, 
which generally suffices to distinguish the true value. 

Let us consider first the most general case. 

(A.) T\co different stars observed at dfferent times. — ^Let 

A, h' = the true altitudes, corrected for refraction, &c., 
T, T' =: the clock times of observation, 
A T, A T' = the corresponding corrections of the clock, 
a, a' = the right ascensions, and 
d, d' = the declinations of the stars at the times of the 

observations respectively, 
tjf = the hour angles of the stars at the times of the 
observations respective!}', 
X = f—t = the difference of the hour angles, 
^ = the latitude of the observer : 

then we have, if the clock is sidereal, 


f= r+ Ar— » 

X =(r— T) + (Ar'— aT)— (a'— a) (804) 

a formula which does not require a knowledge of the absolute 
values of A 7" and a 7"', but only of their difference; that is, of 
the rate of the clock in the interval between the two obser- 

If the clock is regulated to mean time, the interval T' — T+ 
A y — A 7" is to be converted into a sidereal interval by adding 
the acceleration. Art. 49. 

We have next to obtain formulae for determining f and i or (' 


from the data A, A', i, ^', and X. I shall give two general solu- 
tions, the first of which is the one commonly known. 

Mrst SohUion. — ^Let the observed points S and S^ be joined 
by an arc of a great circle SS'. In the triangle PSS^ there are 
given the sides PS= 90^ — 8, PS' = 90° — 3', and the angle SPS' 
= Xj from which we find the third side SS' = -B, and the angle 
PS'S= P, by the formulse [a of Art. 10] 

cos B = sin d' sin d -{- cos d' cos d cos X 
sin B cos P == cos d' sin d — sin ^' cos ^ cos A 
sin J9 sin P e= cos S sin X 

or, adapted for logarithmic computation, 

m sin M= sin d 

m cos M= cos d cos A 

cos 5 = m cos (Jtf" — a') ) (305) 

sin 5 cos P = m sin (M — d^) 
sin 5 sin P = cos d sin A 

In the triangle ZSS' there are now known the three sides 
Z8 = 90*^ — A, jaS' = 90*^ — A', /Sfif' = P, and hence the angle 
ZS'S = Qy by the formula employed in Art. 22 : 

A/\ cos A' sin 5 / ^ ^ 

Now, putting 

ff = P-« 

there are known in the triangle PZS' the sides PS' = 90® — 5', 
^^' = 90*^ — A', and the angle PS'Z = q, from which the side 
PZ= 90*^ — f , and the angle S'PZ= i\ are found by the formul» 

sin f> = sin d^ sin A' + cos ^' cos A' cos q 
cos f> cos ^ = cos d^ sin A' — sin d' cos A' cos q 
cos f sin f = cos A' sin g^ 

or, adapted for logarithmic computation, 

n sin iV = sin A' 

n cos JV= cos A' cos q 
sin sp = n cos (JVT— ^0 ) (307) 

cos 9> cos f = n sin (iV — d') 
cos f sin t' = cos A' sin q 


The fonnulsB (305) and (307) leave no doubt as to the quadrant 
in which the unknown quantities are to be taken. But we may 
take the radical in (306) with either the positive or the negative 
sign, and J §, therefore, either in the first or fourth quadrant. 
If we take J Q always in the first quadrant, the values of q will be 

and either value may be used in (307) ; whence two values of f 
and t'. That value of y, however, which agrees best with the 
known approximate latitude is to be taken as the true value. 
There is also another method of distinguishing which value of q 
will give the true solution ; for, if A^ and A are the azimuths of 
S' and Sj we have in the triangle ZSS' the angle SZS' = A' — A, 

sm Q = sm (A' — A) 

^ ^ ^ sin 5 

in which cos h and sin B are always positive. Hence Q is posi- 
tive or negative according as -4' — A is less or greater than 180°. 
The observer will generally be able to decide this : the only eases 
of doubt will be those where A^ and A are very nearly equal or 
where ^' — ^ is very nearly 180° ; but, as we shall see hereafter, 
these cases arc very unfavorable for the determination of the 
latitude, and are, therefore, always to be avoided in practice 

If the great circle SS^ passes north of the zenith, we shall have 
A^ — A negative, or greater than 180°: hence we have also this 
criterion : we must take q = P — Q or q = P+ Q according as the 
great circle SS^ imsscs north or south of the zenith. 

Second Solution.— Bisect the arc SS', Fig. 25, in T; join FT 
and ZTy and put 

C = ST=:S'T, 

D = the declination of T = 90° — PT, 

I£= the altitude of !r= 90° — ZT, 

T :^ the hour angle of T = ZPT, 
F = the angle PTSy 
Q =-. the angle ZTS, 

q = the angle FTZ. 

We have in the triangle PSS' [Sph. Trig. (25)] 

sin« C = 8in» K(J — S') cos« ) X -]- cos« J (.5 + d') sin* } X 


or, adapted for . logarithmic computation, by introducing an 
auxiliary angle E, 

sin (7 sin j& = sin i (^ — ^') cos J A | ^308^ 

sin (7 cos J& = cos i (^ + a') sin M I ^ ^ 

In the triangle SPT we have the angle PTS = Pj and there- 
fore in the triangle S'PTwe have the angle PTS' = 180° — P, 
the cosine of which will be = — cos P: hence, from these 
triangles we have the equations 


sin D cos + cos D sin C cos P = sin ^ 
sin J) cos G — cos jD sin (7 cos P = sin d' 

2 sin D cos C=Bin d -\- sin d' 
2 cos jD sin G cos P = sin ^ — sin d' 

. « sin i (^ + a') cos H^ — ^0 
sm 2> = ^ — ■ ^ ^^ 

cos C 

^ cos i (^ + ^') sin i (^ — ^') 
cos P = ^^ — ' ^ ^^ 

cos 2) sin G 


which determine D and P after Chas been found from (808). 

In precisely the same manner we derive from the triangles 
ZTS and ZTS' the equations 

„ sin Hh + A') cos J (A — h') 

sm J? = ^^ — ■ — - — - — ^^ 

cos G 

^^ cos i (A + h') sin t (A ~ y) 

cos H sin (7 


Then in the triangle PTZ we have the angle PTZy by the 

q = P-Q 

and h«nce the equations 

sin ^ = sin jD sin ff -{- cos D cos ff cos jr 
cos f cos T = cos jD sin -ET — sin D cos jH" cos q 
cos ^ sin r = cos Jff sin q 


To adapt these for logarithmic computation, let fi and f be deter- 
mined by the conditions* 

cos /9 sin ^ = cos H cos q 
cos /9 cos ^ = sin Jff \ (311) 

sin fi = cos JS'sin q 

then f and r are found by the equations 

sin ^ = cos p sin (D + y) 
cos 9> cos T = cos /9 cos (D + y) Y (^12) 

cos ^ sin r = sin ^ 

To find the hour angles t and t'y let 

a:=:T — Ki' + O 

then, since J A = J (<'—<), we have 

}>l + a: = T — ^ = the angle TPS, 
iX—x = f —T = the angle TPS\ 

and from the triangles PTS and PTS' we have 

sin a X-^x) sin P sin (} A — x) sin P 

sin C cos d sin (7 cos d' 


sin (} >l -|- a:) — sin (i X — x) cos d' — cos d 

sin a ^ + x) + sin (J A — x) cos ^' + cos^ 

and, consequently, 

tan a; = tan J (^ + d') tan } (^ — d') tan } .1 (818) 

Hence, finally. 

As in the first solution, the value of q will become = P+Q 
when the arc joining the observed places of the stars passes north 
of the zenith. 

Example. — 1856 March 5, in the assumed Latitude 39® 17' N. 
and Longitude 5* 6" 36* W., suppose the following altitudes 

* The equations (811) can always be satisfied, sinoe the sum of their squares giTet 
the identical equation 1 = 1. 


(already corrected for refraction) to have been obtained; the 
time being noted by a mean solar chronometer whose daily rate 
was 10'. 32 lodng. The star Arciurus was not far from the prime 
vertical east of the meridian ; Spica was near the meridian. 

Arciurus, h = 18^ 6' 30" Chronometor T = 9* 40« 24'.8 

Spica, K = 40 4 43 " T = 14 38 16.7 

T^T= 4 57 51.9 
d = + 19° 55' 44".6 Corr. for rate = +2 .1 

^' = — 10 24 39 .5 Bed. to aid. time = +48.9 

Sid. interval = 4 58 42.9 
a = 14» 9- 6'.79 a — a' = 51 29 .1 

o! = 13 17 37 .72 k = b 50 12 .0 

= 87^33' 0". 

According to omv first soluiiony we obtain, 

by (305), B = 91° 15' 52".5 P = 69° 57' 54".7 

and, by (306), Q = 64 51 24 .8 

wbonco J = 5 6 29 .9 

Then, by (307), we find 

V' = 39°17'20" <' = 5°3'0"= 0*20"12». 

a' =13 17 37.72 

Sidereal time of the observation of Spica = 13 37 49 .72 
Sidereal time at mean Greenwich noon =: 22 53 39 .83 

14 44 9 .89 

Acceleration for 14» 44- 9'.89 = — 2 24.85 

" longitude ==— 50.23 

Moan time of the observation of Spica = 14 40 54 .81 
Chronometer correction at that time, a T' = -|- 2"' 38*.ll 

According to the second solution^ we prepare the quantities 

}il=43°46'30" \{d+d')= 4°45'32".6 1(A + A')= 29° 5'36".5 

J (5-^') = 15 10 12.1 }(A— A') =— 10 59 6.5 

with which we find, by (308), (309), and (310), 

log tan ^ = 9.437854 D= 6° 34' 32".0 

log sin C = 9.854225 P = 68 27 22 .2 

log cos C = 9.844639 Q = 108 35 12 .1 

log sin I£=^r 9.834176 3 = — 40 7 49 .9 
log cos //= 9.863785 


(The auxiliaries C and H are not themselves required : we take 
their cosines from the table, employing the sines as argomenta.) 
We now find, by (811), (812), (818), and (814), 

log sin fi == n9.673029 t = 322<> 30' 61".8 

log cos fi = 9.946682 x = 1 14 21 .3 

r = 39^ 18' 4".0 T — a; == 321 16 30 
2) + ;- = 46 52 36 .0 = 21* 26- 6* 

9^ = 39 17 20 . iil = 2 65 6 

^ = 18 30 

f= 20 12 

agreeing precisely with the results of the first solution, 

179. In the observation of lunar distances, as we shall see 
hereafter, the altitudes of the moon and a star are observed at 
the same instant with the distance of the objects. The ob- 
served distance is reduced to the true geocentric distance, which 
is the arc B of the ohove first solution^ or 2 C of the second. The 
observation of a lunar distance with the altitudes of the objects 
furnishes, therefore, the data for finding the latitude, the local 
time, and the longitude. 

180. (B.) A fixed star observed at two different times, — ^In this case 
the declination is the same at both observations, and the general 
formulse of the preceding articles take much more simple forms. 
The hour angle k is in this case merely the elapsed sidereal time 
between the observations, the formula (304), since a = a', 
becoming here 

>l = (T'— T) + (aT'— aT) (315) 

Putting d' for d in (308) and (309), we find JE?= 0, cos P= 0, 
P= 90°; and Cand D are found by the equations 

sin C = cos ^ sin } X. sin D = f816) 

cos a ^ ^ 

Since we have q=^ P— Q = 90° — §, the last two equations of 
(311) give 

sin fi = cos H cos (?, cos y = %m H sec /9 


which, by (310), become* 

.a COS } (A + A') sin i (A — h') 
sin p = ^^ — ■ ' ^^ ^ 

sin G 

sin i(h + A') cos J (^ — ^') 
cos y = ^ ' "^ ^ ^ 


cos fi COS C 

Then f and r are found by (312), or rather by the following : 

sin ^ = cos fi sin (D + /) 

tan/9 8in/9 } (318) 

tan T = or sin t = ^ 

cos (D + r) ^^^ 9 

The hour angles at the two observations are 

\Z\~^\\ } (319) 

Here y^ being determined by its cosine, may be either a posi- 
tive or a negative angle, and we obtain two values of the latitude 
by taking either D-\-'( ox J> — 7^ m (318). The first value will 
be taken when the great circle joining the two positions of the 
star passes north of the zenith ; the second, when it passes south 
of the zenith. 

The solution may be slightly varied by finding D by the 

cos J >l ^ ^ 

obtained directly from the triangle TTS (Fig. 25), which is right- 
angled at T when the declinations are equal. We can then dis- 
pense with C by writing the formulae (317) as follows : 

. ^ cos J (^ + ^') sin J (A — A') 
sm /9 = ^^ — ' ^^ ^ 

cos d sin i A 

; (321) 

sin J (^ + ^') cos } (A — A') sin D 

cos Y = ^^-^^^ — ' — ^ 

cos ^ sin b 

* The formulas (315), (316), and (817) are essentially the same as those first 
given for this case by M. Caillet in his Manuel du Navi^atewr^ Nantes, 1818. 


181. (C.) The sun observed at (wo different times. — ^In this case 
the hour angle X is the elapsed apparent solar time. If then the 
times T and T' are observed by a mean solar chronometer, and 
the equation of time at the two obser\'ations is e and e' (positive 
when additive to apparent time), we have 

>l = (T'— T) + (Ar— Ar) — (f'— (822) 

Taking then the declinations d and 5' for the two times of obser- 
vation, we can proceed by the general methods of Art. 178. 

But, as the declinations differ very little, we can assume their 
mean — i.e. the declination at the middle instant between thc^ 
observations — as a constant declination, and compute at least an 
approximate value of the latitude by the simple formulae for a 
fixed star in the preceding article. We can, however, readily 
correct the resulting latitude for the error of this assumption. 
To obtain the correction, we recur to the rigorous fomiulie of our 
second solution in Art. 178. The change of the sun's declination 
being never greater than 1' per hour, we may put cos |(J — 3^ 
= 1. Making this substitution in (308) and (309), and putting 8 
for J (5 — o') so that d will signify the mean of the declinations^ 
and A J for J (d' — d) so that sS will signify one-half the increase 
of the sun's declination from the first to the second observation 
(positive when the sun is moving northward), we shall have 

aJ = — H^ — ^') 
sin A^ 

tan ji&= — 

cos o tan } A 

But Au diminishes with L so that £ always remains a small 
quantity of the same onler as ao ; and hence we may also put 
cos jET— 1. Thus the se<.'v>nd equation of ^oOS) gives 

siu C = cos o sin } i 

and the first of \oOi>) 

. sin a 

which are the same as i-^l*»u as iri^en for the case where the 


declination i< ab>^>Iutolv invuriaWc. Ilonce :vo sensible error i* 
prvHiiU'cd in the values of (.'and D by the us<' of the mean de- 


clination. But by the second equation of (309) P will no longer 
be exactly 90° : if then we put 

P = go** + aP 

we have, by that equation, 

cos dmn^d sin Ad 

sin aP = 

cos D sin C cos D sin i il 
or simply 

cos D %\xi\k 

Then, since q^=P — §, we have 

3 = 90O — C + aP 

The rigorous formula for the latitude is 

sin ^ = sin D sin H -|- cos D cos Jff cos q 

in which when we use the mean declination we take q = 90° — 
Q: therefore, to find the correction of ^ corresponding to a cor- 
rection of y = aP, we have by differentiating this equation, D 
and jBT being invariable, 

cos ^.A^ = — COS D COS JS'sin q,AP 

Ad cos H cos Q 

sin } X 

We have found in the preceding article sin fi = cos JBT cos Q; 
and hence 

A^ = ^l^ILL^ (323) 

COS ^ sm } A 

In the case of the sun, therefore, we conapute the approximate 
ktitude f by the formula (316), (317), and (318), employing for 8 
the mean declination. We then find Ay> by (323) (which in- 
volves very little additional labor, since the logarithms of sin /9 
and sin J A have already occurred in the previous computation), 
and then we have the true latitude 

If we wish also to correct the hour angle r found by this 
method, we have, from the second equation of (47) applied to 


the triangle PTZ (taking b and c to denote the sides FT and 
Z Tj which are here constant), 

cos H cos A 

At = 


cos <p 

in which A is the azimuth of the point T. But we have in the 
triangle PTZ 

cos H cos A = sin f> cos D cos t — cos ^ sin Z> 
Substituting this and the value of aP, we have 

A^ -rv 

At = (tan cos T — tan D) 

sinU ^ -^ 

and, substituting the value of tan D (320), 


A<J / , tan ^ \ 
= I tan f cos T I 

sin i A \ cos \ki 

When this correction is added to r, we have the value that wonld 
be found by the rigorous formulae, and we have then to apply 
also the correction z according to (814). In the present case we 
have, by (313), 

re = — A^ tan d tan } X 

and the complete formulae for the hour angles / and V become 

t =t-|-at — X — \k 

f = T + Ar — X + JA 


y = At — X 

we find, by substituting the above values of at and x, 

^^^ /tan^^C08r__tan£\ 
" \ iiti\k tan J -I ' 

and then wc have 

t =T+y— U 

<'=f + y + J^ 

I (825) 

The corrections Ay> and y are computed with sufficient accu- 
racy with four-place logarithms, and, therefore, add but little to 
the labor of the computation. Nevertheless, when both latitude 
and time are required with the greatest j)reei8ion, it will be pre- 
ferable to compute by the rigorous fonnulre. 


Example.— 1856 March 10, in Lat. 24^ K, Long. 80^ W., 
suppose we obtain two altitudes of the sun as follows, all correc- 
tions being applied : find the latitude. 

At app. time 0* 30" h = 61^ 11' 38";3 (^ ) = — 3^ 61' 62".8 

« 4 30 A' = 18 46 35 .8 (3')= — 3 47 57 .4 

iX= 2* 0" i(h + h') = S9 59 7.1 ^=— 3~"49 55 .1 
= 30<> 0' * (A — A') = 21 12 31 .3 a^ = + 1' 57".7 

The following is the form of computation by the formulie 
(316), (317), and (318), employed by Bowditch in his Practical 
Navigator^ the reciprocals of the equations (316) and of the 
second of (317) being used to avoid taking arithmetical comple- 
ments. This form is convenient when the tables give the secants 
and cosecants, as is usual in nautical works. 

eoseo i X 0.801080 

sec 6 0.000972 coseo nl.l76024 

cosec C 0.802002 cos 9.987854 cos 9.987864 

cos } (* + *') «.884847 cosec 0.192066 2> = — 4o 26' 21^.8 cosec fil.112878 
sin J (A — A') 9.668428 sec 0.080469 

rini? 9.744777 cos 9.919829 cos 9.919829 

sec 0.080207 y= 88 46 88 .0 

/> + y == 29 20 16 .7 sin 9.690161 

= 24® 2' 28".2 sin 9.609990 

K the approximate latitude had not been given, we might also 
have taken D — r = — 38° 10' 69''.3, and then we should have 

C08/9 9.919829 

sin (P — r) n9.791113 

f^ = — 30^ 55' 44".3 sin ip n9.7To942 

To correct the first value of the latitude for the change of 
declination, we have, by (323), 

log A^ 2.0708 
sin p 9.7448 
cosec } X 0.3010 
Becs«> 0.0394 
Af = — 143".2 log Ckf n2.1560 

and hence the true latitude is 

SP' = 24^' 2' 23".2 — 2' 28".2 = 24o 0' 0" 


which agrees exactly with the value compnted by the rigorond 

The approximate time is found by the last equation of (318) 
with but one new logarithm : we have 

Bin /9 9.744777 
cosf' 9.960596 

T = 37° 28' 23" Bin r 9.784181 

By (824) and (325), we find 

log A^ 2.0708 log A^ 2.0708 

cosec i X 0.3010 cot i X 0.2386 

tan ^ 9.6494 tan d yi8.8259 

cos T 9.8996 — 13".7 nl.l353 

+ 83".3 1.9208 

y = + 83".3 — (— 13".7) = + 97" 

T + y = 37°30' 0" = 2»30-0' 

t= 0* 30* 0* r = 4* 30« 0- 

which are perfectly exact. 

. 182. (D.) Tico equal altitudes of the same siar^ or of the sitn. — ^This 
case is a verj' useful one in practice with the sextant when equal 
altitudes have been taken for determining the time by tlie method 
of Art. 140. By putting /i' = A in the formulae (817), we find 
sin ^ = 0, cos /3 =^ 1, and hence (318) gives sin ^ = sin (D + y\ or 
f=D + Y. We have, therefore, for this case, by (320) and (321), 

_ tan d sin h sin D 
tan 2) := cos y = ; 

cos i X sin d i (qo^\ 

which is the method of Art. 164 applied as proposed in Art. 165. 
We give y the double sign, and obtahi two values of the latitude, 
for the reasons already stated. 

The time will be most conveniently found by Art. 140. The 
method there given is, however, embraced in the solution of the 
present problem. For, since we here have sin ^ = 0, we also 
have r = 0, and the hour angles given by (325) become 

t =y-iX 
f = y + iX 


the mean of which giveg 

i(t + f) — y = 

that is, — y is the correction of the mean of the times of obser- 
vation to obtain the time of apparent noon = 0*. This correction 
was denoted in Art. 140 hy ^T^; and since cos r = 0, the formula 
(324) gives, when divided by 15 to reduce it to seconds of time, 

^ Ad tan ^ Ad tan d 

15 sin iX 15 tan i X 

which is identical with (262). Thus it appears that (262) is but a 
particular case of the formula (324), which I suppose to be new. 
The latitude found by (326) will require no correction, since 
sin ^ = 0, and therefore A(p = 0, 

NoTK. — The preceding solutions of the problem of finding the latitude and 
time by two altitudes leare nothing to be desired on the score of completeness and 
accuracy ; but some brief approximatire methods, used only by navigators, will be 
treated of among the methods of finding the latitude at sea, and in Chapter VIII. 

183. I proceed to discuss the effect of errors in the data upon 
the latitude and time determined by two altitudes, and hence 
also the conditions most favorable to accuracy. 

Errors of altitude. — Since the hour angles t and <' are connected 
by the relation <' = < + ^j the errors of altitude produce the same 
errors in both ; for, k being correct, we have dt^ = dt ; and for 
either of these we may also put rfr, since we have, in the second 
general solution of Art. 178, T — x = i{t + <'), and x is not 
affected by errors of altitude. Now, we have the general relations 

sin h = sin ^ sin d -\- cos f cos d cos t 1 /q97\ 

sin A' = sin ^ sin ^' -f cos ^ cos ^'cos t' ) 

which, by differentiation relatively to A, ^ f , and ij give [see 
equations (51)] 

rf A = — cos Ad^ — cos ^ sin A dr 
dh' = — cos A'df — cos f sin A'dr 

in which A and J.' denote the azimuths of the two stars, or of 
the same star at the two observations. 


Eliminating dz and dtp successively, we find 

, sin A' ,, . sin A ,,, 

^ sin (A! — A) ^ sin (A' — A) 

. cos A' ,- cos J. ... 

cos (pdr = -: dh ; dh' 

^ sin (A' — A) sin (A' — A) 


These equations show that, in order to reduce the effect of errors 
of altitude as much as possible, we must make sin {A' — A) ss 
great as possible, and hence A' — A, the difference of the azi- 
muths, should be as nearly 90° as possible. 
If we suppose A' — A = 90°, we have 

d^ = — sin A'dh + smAdh' 
cos <pdr = cos A' dh — co& A dh' 

and, under the same supposition, if one of the altitudes is near 
the meridian the other will be near the prime vertical. If, then, 
the altitude h is near the meridian, sin A will be small while 
sin A' is nearly unity, and the error d^ will depend chiefly on 
the term sin A'dh. At the same time, cos A will be nearly unity 
and cos A' small, so that the error rfr will depend chiefly on the 
term co& A dh^. Hence the accuracy of the resulting latitude will 
depend chiefly upon that of the altitude near the meridian ; and 
the accuracy of the time chiefly upon that of the altitude near 
the prime vertical. 

If the observations are taken upon the same star observed at 
equal distances from the meridian, we have A' = — Aj and the 
general expressions (328) become 

cos ^c?r = — 

2 cos A 

dh — dh' 
2 sin A 

The most favorable condition for determining both latitude 
and time from equal altitudes is sin A = cos A^ or A = 45°. 

Errors in the observed clock times. — An error in the observed 
time may be resolved into an error of altitude ; for if we say that 
dr is the error of T at the obser\^ation of the altitude A, it 


amounts to saying either that the time T — rfjP corresponds to 
the altitude A, or that T corresponds to the altitude h + rfA, dh 
being the increase of altitude in the interval dT. We maj^ 
therefore, consider the time T as correctly observed while h is in 
error by the quantity — dh. Conversely, we may assume that 
the altitudes are correct while the times are erroneous. The 
discussion of the errors under the latter form, while it can lead 
to no new results, is, nevertheless, sufficiently interesting. We 
have, from the formula (804), 

dX = dT'-'dT 

The general equations (827), upon the supposition that h and h' 
are correct, give 

= — cos Ad^ — cos ^ Bin Adt 

= — cos A'df — cos f sin A' (dt + dX) 

where we put dt + dX for di\ since t' = i + L Eliminating dty we 


cos sin A' sin A „ ^^^^^ 

d0 = dk (829) 

^ 8in(^'— ^) ^ ^ 

Eliminating df^ 

and similarly 

-, sin A' cos A ,, 

or = ; dl 

sin {A' — A) 

-^ sin A cos A' ,, 

(BT = dX 

sin {A* — A) 

But we have from the formula r — x = J (< + <') 

dr=\(dt^ dV) 

and hence 

_^sin(^-+^) jj. 

8in(J.'— ^) 2 ^ ^ 

K we denote the clock correction at the time T by #, we shall 


d^ = dt'^ dT 

Vol. L— 18 


or, if we deduce ?? from the second observation, the rate being 
supposed correct, 

The mean is 

d^ = dT — i(dT+dT') 

Substituting the value of rfr and also dX = dT' — dT^ we find, 
after reduction, 

^^^ sin^cosA- ^y^ ain^-cos^ ^y, 

8in(J.' — il) 8in(A'— ^) ^ ^ 

The conclusions above obtained as to the conditions favorable to 
the accurate determination of either the latitude or the time are, 
evidently, confirmed by the equations (329) and (331). In addi- 
tion, we may observe that if dT' = dT^ we have dip = and 
d& =^ dT: a constant error in noting the clock time produces no 
error in the latitude, but aftects the clock correction by its whole 

Errors of declination. — These errors may also be resolved into 
errors of altitude. By difterentiating the equations (327) rela- 
tively to h and 5, we find 

dk = cos qd9, dh' = cos ^dd' 

in which q and 7' arc the parallactic angles at the two observa- 
tions' respectively. If these values arc substituted in (328), the 
resulting values of dip and dr will be the corrections required in 
tlie latitude and hour angle for the errors dd and dd' ;* and, de- 
noting these corrections by a^ and Ar, we have 

ein-4'co8<7 ,^ . sin il coso^ ,^, 

^ 8in(ii' — ^) ^8ih(.l'— ^) 

cos A* cos a , ^ cos A cos q' , ,. 

cos cp AT = do ^— dd' 

^ sin {A' — A) sin (^' — A) 


If the observation h is on the meridian, and A' on the prime 
vertical, we have a^ = — dd ; and in the same case we have, by 

* The error of a quantity and the correction for this error are too frequently cob- 
founded. They arc numerically equal, but have opposite signs. If a thooM h% 
a — 7, it is too great by z; its error i% -\- x; but the correction to reduce it to its 
true value is — x. 


(828), df=^+ dhy and the total correction of the latitude 
= rfA — dd^ precisely the same as if the meridian observation 
were the only one. Hence there is no advantage in combining 
an observation on the meridian with another remote from it, iii 
the determination of latitude: a simple meridian observation, 
or, still better, a series of circummeridian observations, is always 
preferable if the time is approximately known. 

When the sun is observed and the mean declination is em- 
ployed, putting A^ = J {d' — d)y we have dd = a5, dd^ = — a^, 
and, by (832), 

sin A' cos ^ -|- sin A cos q' 
8in(A'— il) 
which, by substituting 

. ,, sin gr cos d . . sinq' cosd 

sm A' = — ^ sm A = — 

cos f cos ^ 


^^___Bin^+|^ ^^ (333) 

sm (A — A) cos ^ 

This is but another expression of the correction (323). 

If, when the sun is observed, instead of employing the mean 
declination we employ the declination belonging to the greater 
altitude, which we may suppose to be A, we shall have dd = 0, 
d8^ = — 2 a5 ; and, denoting the correction of the latitude in 
this case by a'^ , we have, by (382), 

, 2 sin A cos a* ^ 2 »\nq cos (/ cos d 

aV = ^ . A^ = ^ ^ A^ 

sin (A' — A) sin (A' — A) cos ^ 

Under what conditions will a'^ be numerically less than Af ? 

First If both observations are on the same side of the 
meridian, the condition a'^ < Af gives 

2 sin q cos ^ < sin (j' -|- q) 


2 sin q cos ^r' < sin ^ cos q + cos ^ sin q 

tan q < tan ^ 

which condition is always satisfied when h is the greater altitude. 
Secondly. If the observations are on different sides of the 


meridian, q and q' will have opposite signs, and we shall have, 
numerically, sin (9' — q) instead of sin (9' + q). The condition 
A'f < ^f J then, requires that 

2 sin q cos 5^ < sin 5'' cos q — cos ^ sin q 


tan J < I tan 5'' 

Therefore a <p will be greater than Ay> on/y when the observa- 
tions are on opposite sides of the meridian and tan y > i tan <^. 
In cases where an approximate result suffices, as at sea, and the 
correction a^ is omitted to save computation, it will be expedient 
to employ the declination at the greater altitude, except in the 
single case just mentioned.* But to distinguish this ease with 
accuracy we should have to compute the angles q and q' ; and 
therefore an approximate criterion must suffice. Since the 
parallactic angles increase with the hour angles, we may substi- 
tute for the condition tan g > J tan q' the more simple one 
< > J <', which gives 

^ 2 
or {t and i' being only the numerical values of the hour angles) 

Hence we derive this very simple practical rule : employ the sun's 
declination at the greater altitude, ercejH xchen the time of this altitude 
is farther from noon than the middle time, in which case employ the 
mean declination. 

The greatest error committed under this rule is (nearly) the 
value of A^ in (323), when t ^ t. But since in this case 3/ - /', 
and t -\- t^ = Xy we have r — ^ ^, and therefore sin /S := cos f sin r 
= cos (f sin } ?., This reduces (323) to a^ - - — J a5 sec } i. 
Since X will seldom exceed 6*, Ad will not exceed 3', and the 
maxinnuu error will not exceed I'.G. In most cases the error 
will be under 1', a degree of approximation usually quite suffi- 
cient at sea. Nevertheless, the computation of the correction 
A^ by our formula (323) is so simple that the careful navigator 

* BowDiTCii and navigators generally employ in all cases the mean declination; 
hut the ahoTe discussion proves that, if the cases are not to be distinguished, it will 
be better always to employ the declination at the greater altitude. 


will prefer to employ the mean declination and to obtain the 
exact result by applying this correction in all cases. 


184. Instead of noting the times corresponding to the observed 
altitudes, we may observe the azimuths, both altitude and azi- 
muth being obtained at once by the Altitude and Azimuth 
Instrument or the Universal Instrument. The instrument gives 
the difference of azimuths = X. The computation of the latitude 
and the absolute azimuth A of one of the stars may then be 
performed by the formulre of the preceding method, only inter- 
changing altitudes and declinations and putting 180° — A for U 
When A has been found, we may also find t by the usual methods. 


185. By this method the latitude is found from the declinations 
of the two stars and their hour angles deduced from the times 
of observation, without employing tlw altitude itself, so that the result 
is free from constant errors (of graduation, &c.) of the instrument 
with which the altitude is observed. Let 

8, 0' = the sidoroal times of the observations, 
a, a' = the right ascensions of the stars, 
df d' = the declinations ** " 

t, f = the hour angles " " 

h = the altitude of either star, 

^ = the latitude ; 

then, the hour angles being found by the equations 

we have 

sin A = sin ^ sin d -{- cos ^ cos d cos t 

sin h = Bin ip sin d' -f- cos ^ cos d' cos f 
Prom the difference of these we deduce 

tan <p (sin d' — sin d) = cos d cos t — cos ^' cos f 

= cos a cos t — cos d' cos r 

= ^ (cos ^ — cos d') (cos t + cos f) 
+ ] (cos d + cos d') (cos t — cos f) 


and, by resolving the quantities in parentheses into their fiEtctors, 

tan ^ = tan i (^' + d) cos i(f + t) cos } (f — f) 
+ cot i Id' — d) sin i(f + f) sin iif — f) 


If now we put 

m sin M= sin i(f — t) cot } (^' — a) 

m cos Jlf = cos i(f — t) tan } (d' -{- S) 
we have 

tan ^ = m cos [} (^ + f) ^ 3f] (835) 

The equations (334) determine m and JHf, and then the latitude is 
found by (335) in a very simple manner. 

It is important to determine the conditions which must govern 
the selection of the stars and the time at which they are to be 
observed. For this purpose we differentiate the above expres- 
sions for sin A relatively to tp and /. The error in the hour angles 
is composed of the error of observation and the error of the clock 
correction. If we put 

Tj T' = the observed (sidereal) clock time, 
A T = the clock correction, 
dT = the rate of the clock in a unit of clock time, 

we shall have 

t = T-\- ^T^a, f = T + £iT + dT{T — T) — a' 

Difterentiating these, assuming that the rate of the clock is suffi- 
ciently well known, we have 

dt=^dT+d£iT, df = dT + d£,T 

in which rfJ", dT' arc the errors in the observed times, and (/a T 

the error in the clock correction. The difterential equations arc 


dh = — cos A dtp — cos if 9\n AdT — cos tp B\ii A d^T 

dh = — cos A' dip — cos <p sin A' dT' — cos <p ^\n A' d c^T 

in which A and A' arc the azimuths of the stars. The difference 
of these equations gives 

-'If- = «!?_^ dT-v ?i^^^ d r + gi"_ji: - ^'" ^ rf ^T 

cos^ COS -4— COS A' cos -4 — cos -A' cos^' — cos^ 


The denominator cos A — cos A' is a maximum when one of 
the azimuths is zero and the other 180°, that is, when one of the 
stars is south and the other north of the observer. To satisfy 
this condition as nearly as possible, two stars are to be selected 
the mean of whose declinations is nearly equal to the latitude, 
and the common altitude at which they are to be observed will 
be equal to or but little less than the meridian altitude of the 
star which culminates farthest from the zenith. It is desirable, 
also, that the difference of right ascensions should not be great. 

The coefficient of dcs.T\% equal to — cot \{A! + -4), which is 
zero when \{A* + A) is 90° or 270° : hence, when the observa- 
tions are equally distant from the prime vertical, one north and 
the other south, small errors in the clock correction have no 
sensible effect. 

When the latitude has been found by this method, we may 
determine the whole error of the instrument with which the 
altitude is observed; for either of the fundamental equations 
will give the true altitude, which increased by the refraction will 
be that which a perfect instrument would give. 

186. With the zenith telescope (see Vol. 11.) the two stars 
may be observed at nearly the same zenith distance, the small 
difference of zenith distance being determined by the level and 
the micrometer. The preceding method may still be used by 
correcting the time of one of the observations. If at the ob- 
served times T, T' the zenith distances are ^ and ^', the times 
when the same altitudes would be observed are either 

C — C 
T and r' + 

cos ip sin A! 


T-\ — and T 

cos ip sin A 

where f ' — ^ is given directly by the instrument. With the 
hour angles deduced from these times we may then proceed by 
(834) and (335). 

But it will be still better to compute an approximate latitude 
by the formulae (334) and (335), employing the actually observed 
times, and then to correct this latitude for the difference of 
zenith distance. 



By diiFerentiating the formula 

tan f (sin ^' — sin S) = cos ^ cos t — cos ^' cos t 
relatively to tp and i\ we have 

sec' ip (sin ^' — sin ^) dip = cos ^' sin ^ d^ = sin C flin A' cW 

Ilenee, substituting 

dje = dT = 


COS ^ sin A' 

we find 

df = - 

} (C — C) sin C cos ip 

sin 1 (a' — a) cos 1 (^' + S) 

and the true latitude will be ^ + dtp. 



187. To find both the latitude and the clock correction from the dock 
times when three different stars arrive at the same altitude. 

As in the preceding method, we do not employ the common 
altitude itself; and, as we have one more observation, we can de- 
termine the time as well as the latitude. 
Let 5, aS", S'% Fig. 26, be the three points of the celestial 

sphere, equidistant from the zenith Z, at which 
the stars are observed. Let 

Fig. 26. 

Also, let 

T, T\ T" = the clock times of the observations, 

^T = the clock correction to sidereal time at 

the time T, 
dT = the rate of the clock in a unit of 
clock time, 
o, a', a" =- the right ascensions of the stars, 
dy d\ d*' = the declinations 
fj fy f' ^r the hour angles 
h = the altitude, 
tp = the latitude. 

X =f ^t = SPS', 




then, since the sidereal times of the observations are 


e = T + aT 

S' = T' + £iT + dT (T' —T) 

and the hour angles are 

we have 

X = r— T+dT(r— T) — (a' —a) 
r = T"— T + dxlT'— T) — (a" — a) 

The angles X and X' are thus found jfrom the observed clock 
times, the clock rate, and the right ascensions of the stars. The 
hour angles of the stars being ^ < + A, and t + A', we have 

sin A = sin ^ sin d -)- cos ^ cos d cos t 

sin A = sin ^ sin d' -f ^^^ 9 cos d' cos (t + X) 

sin A = sin «p sin d" -\- cos ^ cos ^" cos (^ + ^') 

^ sin ^" + COS sp COS ^" cos (^ + ^^ 

Subtracting the first of these from the second, we have an equa- 
tion of the same form as that treated in Art. 185, only here we 
have i + X instead of i' ; and hence, by (334), we have 

m sin Jf = sin i X cot } (^' — ^ 1 r337^ 

m cos M = cos } X tan } (^' + ^) / 

and putting 

N=::iX-^M (338) 

we have, by (335), 

tan ^ == m cos (t + N) (339) 

In the same manner, from the first and third observations we 

m' sin Jf' = sin i X' cot i (^" -— d) 
w! cos M* = cos J A' tan i (5" + d) 

N'=\X' — M' (341) 

tan sp = m' cos (^ + N') ' (342) 

I (340) 

The problem is then reduced to the solution of the two equa- 
tions (339) and (342), involving the two unknown quantities 
f and /. K we put 

k cos (^ + iV) = — 


there follows also 

kcoB(t + iV') = -^ 

and these two equations are of the form treated of in PL Trig. 
Art. 179, according to which, if the auxiliary i> is determined by 
the condition 

tan * = -, (343) 


we shall have 

tan p + } (N+ iVT')] = tan (45*» — ^) cot i(N'—N) (844) 

which determines ^, from which the clock correction is found by 
the formula 

The latitude is then found from either (339) or (342).* 
To determine the conditions which shall govern the selection 
of the stars, we have, as in Art. 185, 

- cos A d^ — cos 9> sin A dT — cos ^ sin ji d^T 

• cos A' dip — cos f sin A' dT' — cos ^ sin A' dAT 

dh = — cos A'^d^ — cos ^ sin A"dT" — cos ^ sin A^d^T 

By the elimination of d^T^ we deduce the following: 

(sin A — sin A')dhz=^ sin (^' — ^ ) d^ — cos ^ sin A sin A (dT — dT ) 
(sin A' — sin A") dh = sin \a" — A' ) d^ — cos ^ sin A" sin A' (dT*' — dV ) 
(sin A" — sin ^ ) (/A = sin {A — A") d^ — cos sin A sin A'* \dT — dT*) 

Adding these three equations together, and putting 

2 iT = sin {A' — A)-\- sin {A" — A') + sin (^ — A") 

dh = 
dh = 

we find 

d6 sin A (sin A" — sin A') sin A' (sin A — sin A") 

^ ^ ' dT-\ ^ dr 

cos^ 2K 2K 

sin A" (sin A' — sin ^) ,_,, 

By eliminating d<p from the same three equations, we shall find 

* This simple and elegant solution is due to Gacss, Monatlicht Corre*pondenz^ VoL 
XVIII. p. 287. 


d^T = »^P ^ (<^Q« ^' — CQ8 "^ 1 ^jf I gin -4^ (cos A" — cos A) ^^, 

2K IK 

sin A" (cos -<4 — cos A') ,_,, 

The denominator 2^ is a maximum when the three differences 
of azimuth are each 120°,* which is therefore the most favorable 
condition for determining both the latitude and the time. In 
general, small differences of azimuth are to be avoided. 

Gauss adds the following important practical remarks. It is 
clear that stars whose altitude varies slowly are quite as available 
as those which rise or fall rapidly ; for the essential condition is 
not so much that the precise instant when the star reaches a 
supposed place should be noted, as that at the time which is 
noted the star should not be sensibly distant from that place. 
We may, therefore, without scruple select one of the stars near 
its culmination, or the Pole star at any time, and we can then 
easily satisfy the above condition. Moreover, at least one of the 
stars will always change its altitude rapidly when the condition 
of widely different azimuths is satisfied. 

The stars proper to be observed may be readily selected with 
the aid of an artificial globe, and in general so that the intervals 
of time between the observations shall be so small that irregu- 
larities of the clock or an error in the assumed rate shall not 
have any sensible influence. 

Having selected the stars, the clock times when they severally 
arrive at the assumed altitude are to be computed in advance 
within a minute or two, so that the observer may be ready for 
each observation at the proper time. This is quickly done with 
four-place logarithms by the formula (267), in which (p and ^ 
will have the same values for the three stars. 

• For by patting a=. A' — J, a' = A" — A'^ we have 

2 JST = sin 11 -}- sin a! — sin (a + a') 

and, differentiating with reference to a and a\ the conditions of maximum or mini- 
mam are 

cos a — cos (a -f" <*')== ^ 
cos a' — cos (a -|- a') = 

which give either a :— a' :^ or a = a' = 120®; and the latter evidently belongs to 
the case of maximum. 


If it is desired to compute the differential formulffi, the follow- 
ing form will be convenient We have 

JT = — 2 sin } (^' — A.) sin } (A" — A') sin 1 (A — A") 

d^ __ sin A cos } (A" + A') sin j (A" ^ A') 
15 cos ^ K 

sin A' cos iU + ^'0 sin i(A — A") ^ 

. sin^cos } (A' + ^) sin } (A' - ^) .^, 
+ ^ dT 

, Bin ^' sin ijA + A") sin } (^ — X") _, 
Bin A" Bin i (X' + A) Bin | (A> — .1) ^^„ 

where dip is divided by 15, since it will be expressed in seconds 
of arc, while dT^ dT\ and dT^' are in seconds of time. If we 
first compute the coefficients of the value of d^T^ those of 
df will be found by multiplying the former respectively by 
cot J {A' + A'')y cot l(A + A'% and cot J {A' + A), and also by 
15 cos f. It is well to remark, also, for the purpose of verifica- 
tion, that the sum of the three coefficients in the formula for df 
must be — 0, and the sum of those in the formula for rf^T^must 
be = - 1. 

The substitution of dX for dT-dT, and dX' for dT'—dT, 
will reduce the above expressions to a more simple form, which 
I leave to the reader. 

Example. — To illustrate the above method. Gauss took the 
following observations, with a sextant and mercurial horizon, at 
Gottingen, August 27, 1808. The double altitude on the sextant 
was 105° 18' 55". The time was noted by a sidereal clock 
whose rate was so small as not to require notice. 



a Andromeda T == 21* 88" 26* 
a Urs(B Minoris T = 21 47 80 
aiyroj 2"= 22 6 21 

The apparent places of the stars were as follows : 

a AndromedfB a 
a TJtscr Minoris a! 
a Lyr<B 


28» 58- 33'.83 

55 4.70 

18 30 28.96 

d =28*» 2'14".8 
d' = 88 17 5 .7 
^" = 38 87 6 .6 

Hence we find 

j;i = — 5**18'25".28 
\ (d' — d)= 30 7 25 .45 
i (d' + d)= 58 9 40 .25 

log cot } (^' — ^) 0.2363973 
log sin i X n8.9661070 

log msinM n9.2025043 

log tan } (^' + d) 0.2069831 
log cos U 9.9981343 

log m cos if 0.2050674 

} A' = 44<> 59' 55".28 
i (a" — ^) = 5 17 25 .90 
i (^" + ^) = 88 19 40 .70 

log cot } (^" — d) 1.0333869 
log sin i X' 9.8494751 

log m' sin M' 0.8828620 

log tan 1 (r + d) 9.8179461 
log cos } X' 9.8494949 

log m' cos JBT 9.6674410 

log tan M 
log cos M 
log m 


log tan M' 
log sin M* 

Jf == — 5^0^ 87".96 W = 

Ji— Jf = J\r= + 22 12.68 hX'--M'=ir= 


86*» 3^ 55".07 
41 30 59 .79 


* = 11** 58' 41".28 log ^ = log tan « 9.3235372 

450 _^ ,3= 33 6 18 .72 log tan (45*» — *) 9.8142617 

\{N'—N) = — 20 56 36 .24 log cot J (JV^ — JV) n0.4171063 
t+}(JV^' + JNr) = — 59 35 14 .71 logtanp+}(-ZV''+^^]n0.2313680 
K^ + ^ = — 20 84 23 .56 

t= — 39 51.15 = — 2»36- 3*.41 

a= 23 58 83.33 
^ + a = e = 21 22 29 .92 
T= 21 33 26. 
Clock correction a T = — 10 56 .08 

Then, to find the latitude, we have 


t + N= — 88^ 88' 38".47 / _[. JVT = — 80*» 31' 50".94 

log cos (t + N) 9.8926738 log cos {t + N*) 9.2162110 

logm 0.2072029 log m' 0.8836667 

log tan ^ 0.0998767 log tan ^ 0.0998767 

q> = 51*^ 31' 5r'.46 

If with these results we compute the true altitude of the 
stars, we find from each h = 52° 37' 21".2. The refraction was 
42".7, and hence the apparent altitude = 52° 38' 3".9. The 
double altitude observed was, therefore, 105° 16' 7".8. The 
index correction of the sextant was — 3' 30", and hence the 
double altitude given by the instrument was 105° 15' 25", 
which was, consequently, too small by 43". 

To compute the differential equations, we find 

A = 293*^ 45'.2 A' = 182*^ 9'.1 A" = 90° 17'.9 

and hence 

dip = ^ 3.808 dT— 0.288 <f T' — 3.519 dT' 

db.T= — 0.391 (fT— 0.007 dT — 0.602 dT" 

by which we see that an error of one second in each of the 
times would produce at the most but 7".6 error in the latitude, 
and one second in the clock correction. 

188. Solution of the preceding problem by Cagnoli's fomiidce. — 
After Gauss had published the solution above given, he was 
himself the first to observe* that Cagnoli's formulse for the 
solution of a very difierent problemf might be applied directly 
to this. 
Wlien the altitude is also computed, Cagnoli's formulie have 

slightly the advantage over those of Oauss. To 
Pig. 2«. (6i«).^ deduce them, let 7, 7', 7" be the parallactic angles 

at the three stars, or (Fig. 26) let 

qr=PSZy q' = PS'Z, q'' = PS"Z, 

and also put 

q =iiPS"S' — PS'S") 

Q' = i iPS"S — PSS") 
Q" = i(PS'S —PSS') 

* Monatliche Corretpondenz, Vol. XIX. p. 87. 

t Namely, that of determining, from three heliocentric places of a solar spot, tht 
position of the sun's equator, and the declination of the spot. — See Caoxou*s 
Trigonomitriey p. 488. 


then, aince 2iSS', ZS'S", and ZSS " are isosceles triangles, we 

q + PSS' = PS'S — 4 

^ + PS'S" = P8"S' — g" 

q + PSS" = PS"S — g" 


? + 2' = 2$" 
«'+?" = 2(2 

S" + g =2$' 
« + 3' + «"=«+ <2'+ «" 

i =-Q+Q'+Q" ) 

2'= C -«'+$" k345) 

«" = «+«'-«" J 

Now, Q, $', Q" are found from the triangles P<S'"<S", PS"S, 
and PS'S, by Napier's Analogies (Sph. Trig. Art. 73), thus : 

tan Q = i^ icot } (X' — l) 

C08l(a"+i') ^ ^ 

^ ^, Bin i id" — d) ^ , ,, . 

tanr= '^"*^''~'^ cotM 
COS i (a' + S) 

where ^, ^' are the angles at the pole found as in the preceding 
article. With these values of §, §', §", those of y, g', and j" 
become known by (845). 
We have also 


and firom this 


cos ^ sin (f '\- X^ = cos h sin ^f' 
cos ^ sin ^ = cos h sin q 

sin (^ + ^) sin q[ 

sin f sin q 

sin (^ + >l) + s5^ ^ sin ^ + ^i^ ? 

sin (f + >l) — sin f sin g' — sin q 

tan (e + U) _ tan K^ + g) 
tan 1 A tan } (^ — s^) 


Substituting the values of q and q' in terms of §, this ^ves 
tan (t + iA) = tan ik tan Q" cot (Q — Q') 

or, substituting the value of tan ^', 

tan (< + U) = "" I ^^1 7 2 cot ( e - C) (347) 

COS } (^ + V 

which determines < + J ^, whence i and the clock correction. "We 
can now find the latitude and altitude from any one of the 
triangles PSZ, PS'Z, PS^'Z, by Napier's Analogies (Sph. Trig. 
Art. 80) : thus, from PSZ we have 

tan KsP + K) = ^^iLOHh?) ^^ ,4^0 ^ j^) 

cos i (^ — q) 

tani(^ -h) = ^'^\^l ^ |cot (450 + H) 

sin i (^ + ^) 

andthen^ = J(f + A) + i(f-'i)» A = H?^ + 'i)-J(f-A). . 
As all the angles are determined by their tangents, an am- 
biguity exists as to the semicircle in which they are to be taken; 
but, as Gauss remarks, we may choose arbitrarily (taking, for 
example, (>, (?', Q'^ always less than 90°, positive or negative 
according to the signs of their tangents), and then, according to 
the results, will have in some cases to make the following 
changes : 

1. K the values of tp and h found by (348) are such that 
cos ip and sin h have opposite signs, we must substitute 
180° -r q for q and repeat the computation of these two equa- 
tions. In this repetition the same logarithms will occur as 
before, but difterently placed. 

2. If the values of f and h exceed 90°, we must take their 
supplenieuts to the next multiple of 180°. 

3. The latitude is to be taken as north or south according 
as sin <p and sin Ii have the same or different signs. 

No ambiguity, however, exists in practice as to /+ \K found 
by (347), since Q — §' can differ from its true value only by 
180°, and this difference does not change the sign of cot (§ — Q^)i 
hence tan (t + JA) will come out with its true sign; and between 


the two values of ^+ JJl, differing by 180°, or 12*, the observer 
will be at no loss to choose, as he cannot be uncertain of his 
time by 12*. 

Example. — Taking the example of the preceding article, we 
shall find 

^ == — 37*» 57' 9".3 C' = + 6*» 17' 5r.66 Q"= ~ 84<> 25' 23".81 

J = — C + e' + C" = ~ 40*» 10' 22".85 

e = — 39 51 .27 

i(t + g) = — SG** 35' 37".06 i(t --q) = + O*' 34' 45".79 
i (V + h) = 52 4 36 .35 i (sp — A) = — 32 44 .84 
^ = 51 31 51 .5 h = 52 37 21 .2 

189. If we have observed more than three stars at the same altitude, 
we have more than sufficient data for the determination of the 
latitude ; but by combining all the observations we may obtain 
a more accurate result than from only three. This combination 
is effected by the method of least squares, according to which 
we assume approximate values of the unknown quantities and 
then determine the most probable corrections of these values, or 
those which best satisfy all the observations. 

Let T, T\ T\ T", &c. be the observed times by the clock 
when the several stars reach the same altitude. Let a The the 
assumed clock correction at some assumed epoch = Tq\ 8T\h^ 
known rate. Let <p and h be the assumed approximate values of 
the latitude and altitude. With <p and A, which will be the same 
for all the stars, and with the declinations 5, 5', 5", &c., compute 
the hour angles <, t\ <", &c. and the azimuths -4, A', -4", &c. If 
the assumed values were all correct and the observations perfect, 
we should have a + t= T+ £,T-\' dT{T— T^y.iov each of these 
quantities then represents the sidereal time of observation ; but 
if <p, A, and a T require the corrections dtp^ dh, and cJaT', and if 
dt is the corresponding correction of <, we shall have 

a + ^ + jt= r+ AT + rfAT + arcr— To) 

The relation between dip, dh, and dt is 

dh = — cos Adif — 15 cos <p sin Adt 
and a similar equation of condition exists for each star. In all 

Vol. L— 1» 


these equations, dh and df are the same, but dt is different for 
each. K we put 

f = T + £iT+dT(^T — r.) — (a +t) 

which are all known quantities, we have 

dt=f+d£iT, dt'=f + dAT,&c. 

and the equations of condition become 

dh-{-coBA .d^ -{-15co8^BmA .^AT-[-15co6^8inui ./ =0 
dh -\- coaA^d^ -\- l^cos^ainA' .d^T+lb cos^sin^' ,f =zO 
dh + cos A", d^ + lbcosf sin A", d a T + 15 cos ^ sin A"./" = I (**®) 


from which, by the method of least squares, the most probable 
values of dh, dip, and di^T are determined. The true values of 
the altitude, latitude, and clock correction will then be A + ^'A, 
<p + dip, ^T+d^T. 

The hour angles will be computed most accurately by (269)| 
which is the same as the following : 

tan« i t = ^^" K^ ^ ^ + ^) sin i (C + y — ^) 
cos i (C + f + ^) cos i (C — ^ — ^) 

in which ^ = 90° — h ; and the azimuths by 

tan« 1 A = »^" * ( ^ ~ y + ^) c os KC — y — ^) 
cos J (C + sp + ^) sin i (C + s^ — d) 

Since <p and ^ are constant, it wll be convenient to put 


sin {c -\-\d) sin (ft — j ^) 

m = ; n = ^ 

cos (^ + i ^) cos (c — } ^) 

tan« i ^ = mn tan« M = - (851) 

The barometer and thermometer should be observed with each 


altitude, and if they indicate a sensible change in the refraction 
a correction for this change must be introduced into the equations 
of condition. Thus, if r^ is the refraction for the altitude h for 
the mean height of the barometer and thermometer during the 
whole series, while for one of the stars it is r, then the assumed 
altitude requires for that star not only the correction dA, but also 
the correction r — r^. Hence, if we find the refractions r, r', r", 
&c. for all the observations, and take their mean r^ we have only 
to add to the equations of condition respectively the quantities 
r — r^, r' — r^, r" — r^, &c. 

If any one of the stars is observed at an altitude \ slightly 
different from the common altitude A, we correct the correspond- 
ing equation of condition by adding the quantity h — \. 

190. We may also apply the preceding method to the case 
where there are but three observations. The final equations are 
then nothing more than the three equations of condition them- 
Belves, from which the unknown quantities will be found by 
simple elimination. It will easily be seen that this elimination 
leads to the expressions iordf and d a T' already given on p. 284, 
if we there exchange dT^ dT'^ and dT^' for/,/', and/" respect- 
ively. We can simplify the computation by assuming a T so as 
to make one of the quantities /, /', /" zero. Thus, we shall 
have/= if we determine A^by the formula 

Ar=a + « — [T-j-arcr— ?;)] (352) 

then, finding/' and/" with this value, and putting 

^ sin } A! cos 1 A! , 

"" sin 1 (.i' —A) sin } (A" t- A') ' ^ 

yf 9m\A'' QO%\A" ^ff 

"" sin } (A" — A) sin J (A" — A) 

we shall have the following formulae: 

di^T= — A:' sin J (A + A") -f A" sin \ (ii'+ A) 

-^^ = -^cosK^ + A'0 + ^'cos}(A'+A) ^ (353) 

.^-^ — = + jfc'cos J(A" — A) — A"cos J(ii'-- A) 
15 cos f 


Example. — Taking the three observations above emplojed, 
and assuming the approximate values 

A T = — 11* ()•, ^ = 51^ 32' 0", A = 52<> 37' 0", 
we shall find, by (351), 

t = — 2» 36* 5-.50 t' == — 3» 19* 55*.65 ^ = 3* 23- 58*.25 
A = — 66<> 16'.2 A' = — 177° 50'.2 A" = 90« 18M 

By (349), putting in this case 5 T = 0, we then have 

/ = — 1-.83 /' = + 80-.95 /" = — d'.21 

and the equations of condition (350) become 

dh + 0.4027 dip — 8.5410 <f a T + 15.63 = 
dh — 0.9993 dip — 0.3522 d a T — 28.51 = 
dh — 0.0053 d^ + 9.3308 d£iT— 57.94 = 


d^T= + 3-.92 dip = ^ 8".58 dh = + 21".81 

and the true values of the required quantities are, therefore, 

A T = — 10* 56-.08 f = 51° 31' 51".42 h = 52° 37' 21".81 

agreeing almost perfectly with the values before found. 

Since in this example there are but three observations, we 
may also employ the formulse (353), first assuming 

Ar= — 10-58M7 

which is the value given by (352). With this we find 

/' = + 82'.78 /" = — 4v38 

log ^ = 0.4199 log A" = nO.4932 

and by (353) we shall find 

d^T= + 2'.09 cff = — 8".58 dh = + 21".31 

Hence the true clock correction is — 10* 58M7 + 2'.09 = 
— 10* 56\08; and the values of the latitude and altitude also 
agree with the former values. 


191. We may here observe that, theoretically, the latitude 
might be found also from three different altitudes of the same 
star and the differences of azimuth ; for we should then have 

sin d = sin tp sin h -\- cos fp cos h cos A 

sin ^ = sin ^ sin K + cos ^ cos h' cos (A. -\- k) 

sin ^ = sin f» sin K' + cos ^ cos h" cos {A + k') 

in which A is the azimuth of the star at the first observation, 
and the differences of azimuth X and X' are supposed to be given. 
The solution of Art. 187 may be applied to these equations by 
writing h for d and A for t 

Again, there might be found from three different altitudes of 
the same star not only the latitude and time, but also the decli- 
nation of the star; for we then have 

sin k = sin ^ sin d -f cos ^ cos d cos t 

sin h' = sin ^ sin d -|- cos ^ cos d cos (t + '^ ) 

sin h" = sin f sin ^ -f cos ^ cos d cos (t + X') 

from which we can readily deduce ^, ^ and d. But the method 
is of no practical value, as the errors of observation have too 
much influence upon the result. 



192. We may observe the time of transit of a star over any 
vertical circle with a transit instrument (or with an altitude and 
azimuth instrument, or common theodolite) ; for when the rota- 
tion axis is horizontal, the collimation axis will, as the instru- 
ment revolves, describe the plane of a vertical circle. Tor any 
want of horizontality of the rotation axis, or other defects of 
adjustment, corrections must be applied to the observed time of 
transit over the instrument to reduce it to the time of transit 
over the assumed vertical circle. These corrections will be 
treated of in their proper places in Vol. 11. ; and I shall here 
assume that the observation has been corrected, and gives the 
clock time T of transit over some assumed vertical circle the 
azimuth of which is A. The clock correction a T being known, 
we have the star's hour angle by the formula 


and then, the declination of the star being given, we have the 
equation [from (14)] 

COS t sin f» — tan d cos f» = sin t cot A (354) 

If, then, A is also known, the latitude ip can be found by thia 
equation. Let us inquire under what conditions an accurate 
result is to be expected by this method. By differentiating the 
equation, we find [see (51)] 

cos fl' cos ^ .^ tan C , . . sin at ,^ 

dip = at a A -\ — ad 

cos C sin A sin A cos C sin A 

from which it appears that sin A and cos Q must be as great as 
possible. The most favorable case is, therefore, that in which 
the assumed vertical circle is the prime verHccd^ and the star a 
declination differs but little from the latitude ; for we then have 
A = 90° and (^ small. Indeed, these conditions not only increase 
the denominator of the coefficient of rf/, but also diminish its 
numerator, since, by (10), we have 

cos q cos ^ = sin C sin ^ 4~ ^^ ^ ^^^ 9 ^^^ ^ 

which vanishes wholly when the star passes through the zenith. 
Moreover, if the same star is observed at both its east and we:5t 
transits over the prime vertical, we shall have at one transit sin 
A = — 1, at the other sin J. == + 1, and the mean of the two 
resulting values of the latitude will, therefore, be wholly free 
from the effect of a constant error in the clock times, that is, of 
an error in the clock correction. It is then necessary only tliat 
the rate should be knowTi. This method, therefore, admits of a 
higli degree of precision, and requires for its successful applica- 
tion only a transit instrument, of moderate dimensions, and a 
time-piece. Its advantages were first clearly demonstrated by 
Bessel* in the year 1824 ; but it appears that verj' early in the 
last century Romer had mounted a transit instrument in the 
prime vertical for the purpose of determining the declinations of 
stars from their transits, the latitude being given. The details 
of this important method will be given in Vol. 11., under 
*' Transit Instrument.*' 

* Attronom, iVocA., Vol. IH. p. 9. 


193. It may sometimes be possible to observe transits only over 
some vertical circle the azimuth of which is undetermined. We 
must then observe either two stars, or the same star on opposite 
sides of the meridian. We shall then have the two equations 

cos t . tan A sin ^ — tan d . tan A cos f> = sin f 
cos f . tan ii sin ^ — tan d^ . tan A cos f = sin ^ 

from which the two unknown quantities A and f can be deter- 
mined. K the same star is observed, we shall only have to put 
d' = d. Regarding tan A sin f and tan A cos f as the unknown 
quantities, we have, by eliminating them in succession, 

- . sin t sin d' cos d — sin t' cos d' sin d 
tan ii sin ^ = 

cos t sin ^' cos d — cos f cos d' sin d 

^ , — sin (f — t) cos d' cos d 
tan A COB^ = ^^ 

cos t sin d' cos d — cos f cos d' si 

sin d 
Jf we introduce the auidliaries m and Jf, such that 

m sin M= sin (d' + d) sin } (f — t) 
m cos M= sin (a' — d) cos i(f — t) 

} (355) 

we shall easily find 

m sin [} (^ + — -*n = 81^ * 81^ ^' c^8 9 — sin ^ cos d' sin 9 
m cos [} (^ + — -^ = c^s * si"^ ^' cos d — cos f cos d' sin J 
m sin [} (f — t) — Jf] = — sin (t' — t) cos d^ sin ^ 

and hence 

tan il sin f = tan {_i'(f + t) — Jf] \ 

, , sin[J(^— — -^cot^ / (^^^) 

tan Acos0 = 5i_i^ ^ =* V 

cos [J («' + ^) — Jf] j 

which determine A and f> by a simple logarithmic computation. 
The solution will be still more convenient in the following form : 

^ Bin («' — ») 

tan a» = tan d "-^ ^ ^ =L > r 357) 

8in[i(<' — — J"] 

^^ ^ _ tan[J(f + 0-^ 

sin f 


If the same star is observed at each of its transits over the 
same vertical circle, we have i' = ^, and hence tan Jif=QOy 

JIf == 90°, which gives 

tau f = tan d ^^ — ' — - tan A = ^^ — ■ — ^ (358) 

cos i (^ — ^^^ 9 

K the same star is observed twice on the prime vertical, we 
must have <' + < = 0, since tan ^ = oo ; and then, 

. tan d tan d ,^.^. 

tan f = = (859) 

cos i (<' — t) cos t 

which follows also from (354) when cot ^ = ; or, geometrically, 
from the right triangle formed by the zenith, the pole, and the 
star, as in Art. 19. 

If the latitude is given, we can find the time from the transits 
of two stars over any (undetermined) vertical circle by the second 
equation of (357), which gives 

sin [}(«'+ — ^ = -^^^^ w° [i (^ — — -^] 

tan d 

for the observation furnishes the elapsed time, and hence V — i\ 
and this equation determines \{V + t\ and hence both i and f. 

If the latitude and time are given, \ye can find the declination 
of a star observed twice on the same vertical circle, by (358). 
When the observ^ation is made in the prime vertical, this becomes 
one of the most perfect methods of determining declinations. 
See Vol. n., TVansit Instrument in the Prime Vertical. 

194. The following brief approximative methods of deter- 
mining the latitude may be found useful in certain cases. 



195. (A.) By tico altitudes near the meridian and the chronometer 

times of the observations^ when the rate of the chronometer is known, 

but not its correction. 


h, K = the true altitudes, 

Tj T* ^^ the chronometer times, 


then, t and t' being the (unknown) hour angles of the observations, 
we have, by (287), approximately, 

h^ = h' + af* 

in which h^ is the meridian altitude, and 

225 sin 1" cos cos d 
a = ^ 

2 cos \ 

The mean of these equations is 

and their diflference gives 

But we have 

T = i(!r'— T) = i(f—t) 

in which we suppose the interval T' — Tto be corrected for the 
rate of the chronometer. Hence 

2 ar 

which, substituted in the above expression for h^, gives 

h, = i{h + h') + ar« + [^(^-"^')]' (360) 

According to this formula, the mean of the two altitudes is 
reduced to the meridian by adding two corrections: 1st, the 
quantity ar*, which is nothing more than the common "reduc- 
tion to the meridian*' computed with the half elapsed time as the 
hour angle ; 2d, the square of one-fourth the diflerence of the 
altitudes divided by the first correction. 

If we employ the form (285) for the reduction, we have 

in which 

h^ = i(h + A') + Am + ^^^^ ^ ^'^^' (361) 


. cos f cos d 2 sin' }t 

A ■== m = 

cos h^ sin 1" 

and m is taken from Table V. or log m from Table VI. 


Example 1. — ^From the observations in the example of Art 
171, 1 select the following, which are very near the meridian. 

Obsd. alts. Q 

True alts. © 


50° 5' 42".8 h' = 

50^21' 7".6 

23* 50- 4«'.5 

50 7 25 .5 h = 

50 22 50 .4 




T= 4 55.5 

Hh + h') = 

50 21 59 .0 

Am — 

+ 59 .0 

log m 1.6778 

2d corr. = 

+ 11 .2 

log A 0.0930 

A. = 

50 23 9.2 

log Am 1.7708 

c, = 

39 36 50 .8 

log[j(A — V)]« 2.8198 

i, - 

- 1 48 9 .2 

log 2d corr. 1.0490 

f = 37 48 41 .6 

Example 2. — In the same example, the first and last observa- 
tions, which are quite remote from the meridian, are as follows: 

Obsd. alts. Q True alts. Q Chronometer. 

49<> 51' 19".3 h = 50<> 6' 43".7 23» 37- 35- 

49 50 24 h' = 50 5 48 .4 18 31 

J(A — A')= 13 .8 T= 20 28 

which give Am = 16' 58", and the 2d corr. = 0".2, whence 
^ = 37° 48' 37". 

This simple approximative method may frequently be useful 
to the traveller, and especially at sea, where the meridian obser- 
vation has been lost in consequence of flying clouds. At sea, 
however, the computation need not be carried out so minutely 
as the above, and the method becomes even more simple. See 
Art. 204. 

M. V. Caillet* gives a method for the same purpose, which is 
readily deduced from the above. Put 

A =A' — A t'= r— T=2r 

then (360) becomes 

' 2 4 4aT" 

* TraM de Navigation (2d edition, Paris, 1S57), p. 819. 


or, patting 

sin 1" 

in which h is the altitude farthest from the meridian. Although 
this reduces the two corrections of (361) to a single one, the 
computation is not quite so simple. 

196. (B.) By three altitudes near the meridian and the chronometer 
times of the observationSy when neither the correction nor the rate of the 
chronometer is knmon. — In this case we assume only that the chro- 
nometer goes uniformly during the time occupied by the observa- 
tions. Let 

A, A', h" = the true altitudes, 

T, T'f T" = the chronometer times, 

Tj =: the chronometer time of the greatest altitude. 

If we introduce the factor for rate = A, according to Art. 171, 

the formula for the reduction to the meridian by Gauss's method 

is, approximately, 

h^ = h + aki* 

in which t is the time reckoned from the greatest altitude. De- 
noting ak by a, we have then, from the three observations, 

\ = h'+aiT — T^y y (363) 

A^ = A"+a(T"-T,)« j 

which three equations suffice to determine the three unknown 
quantities a, T^, and hy By subtracting the second from the 
first, and the third from the second, we obtain 


= a{T + T) — 2o7; 

A' A" 

- =a(r'-f T') — 2ar, 

rpn rp 

and the difference of these is 

A' — A" A — A' 

rptf rpf rpt rp 

= a{T"— T) 


If, then, we put 

b = — = the mean change of altitude in one second 

of the chronometer from the first to the 
second observation, 

A' — - A" 

^ = ■;;;:; 7^. ^= ditto from the second to the third obser- 

fpff fpf 

we have 

c — b 

rpn rp 

^^ T+r b ^ ^ T+ T" c 

* 2 2a ' 2 2a 


Having thus found T^, we can find h^ from any one of the equa- 
tions (363), all of which will give the same result if the compu- 
tation is correct.* 

Example. — Prom the observations in the example of Art 171 
I select the following three observations : 

Obsd. alts. Q^ True alts. Chronometer. 

50°5'42".8 h =60^21' 7".6 T = 23* 50- 46'.5 

50 7 27 . A' = 50 22 51 .9 T' = 23 55 16 . 

50 7 25 .5 h" = 50 22 50 .4 T" = 37 .5 

A — A' = -- 104".3 T' — r = 269-.5 6 = — 0.8869 

A' — A" = + 1 .5 T" —T= 321 .5 c = + 0.0047 

T"— T=591. c — fe = + 0.3916 

Kr+r') = 23.53. 1..3 log. =7^8 

— — = + 4 52.0 log (T — T^y = 5.2604 


T^ ~ 23 57 53 .3 
7; — 7- 6*.8 


ioga(r r,y = 2.0817 

A — 50° 21' 7".6 

T,y — + 2 .7 

A, — 50 23 8 .3 

Cj — 39 36 51 .7 
^, — 1 48 9 .2 

y = 37 48 42 .5 

The moan of the throe values found from these altitudes in Art 
172 is 37° 48' 42".8. 

* This method is estsentially the same as that proposed bj Littbow (Attrommnt, 
Vol. I. p. 171.) 1 hare here rendered it applicable to the sun without considering 
the change of declination, by introducing Gausses form for the redaction to the 


197. (C.) By iioo altitudes or zenith distances near the meridian 
and the difference of the azimuths. — If the observer has no chrono- 
meter, he may still obtain his latitude by circummeridian alti- 
tudes, if he observes the altitudes with a universal instrument, 
and reads the horizontal circle at each observation, taking care, 
of course, that the star is always observed at the middle vertical 
thread. As this instrument generally gives directly the zenith 
distances, we shall substitute (^ for 90° — h. We have the equa- 

sin d = sin ^ cos C — cos f sin C cos A 
^ sin (f — C) + 2 cos ^ sin C sin'Jil 


cos } (sp + ^ — C) sin } [C — (^ — ^)] = cos f sin C sin* } A 


f — ^ = Ci = the meridian zenith distance; 

and hence 

. , ,_ -,. cos f sin Cain'} il ,^--^ 

sm } (C — Ci) = — (865) 

'^ ^^^ COS [^ - i(C - O] 

which expresses the reduction to the meridian = {^ — {^j when 
the absolute azimuth A is given. If the observation is very 
near the meridian, we may neglect J (^ — ^j) in the denominator 
of the second member, and take 

cos f sin Ci 2 sin* \ A 

* cos d sin 1" 

or, putting 

cos f sin Ci sin 1'' 

*i— ^ ■ ■ ■ ■ ■ ■■ ■ • w^^_^^^^Has 

« = " : ' • -^ (366) 

cos d 2 

C — Ci = aA^ (367) 

from which it follows that near the meridian the zenith distance 
varies as the square of the azimuth. 
Now, when we have taken two observations, we have 

C, = C — aA^ 
Z,= t; —aA!' 

whence, putting 


we deduce tlie following equation, analogous to (360), 

Here r is equal to one-half the difference of the readings of the 
horizontal circle, and is therefore known ; and the computation 
is entirely similar to that of the formula (360). 

198. (D.) By three altitudes or zenith distances near the meriditBH 
and the differences of azimuths. 

Supposing the observations taken with a universal instra- 
ment, let 

C, C, C = the true zenith distances, 
A, A', A" = the readings of the horizontal circle, 

we shall have, by the preceding article, 

C,= C -a(A -^)« ^ 

C,= r -a(A' ^A,y I (369) 

c, = r - a (A" - A,y j 

in which A^ is the (unknown) circle reading in the meridian, 
and a is the (unknown) change of zenith distance for 1" of azi- 
muth. These equations are solved in the same manner as (363); 
and hence we have the formulae 

= c = 

A' — A A" — A' 

c — h 

a = 

A'' — A 

A + A' h , A' + A" c 
A = — ! or A- = ' 

* 2 2a * 2 2a 


which determine a and Ap after which {^jis found by any one of 
the equations (369).* 

* In this connection, see an article by Littsow in Zaoh*8 MotuUUcke CorrttpomiaUf 
Vol. X. (1824). 




199. We have, Art 149, 


= cos ^ sin A 

If then we observe two altitudes near the prime vertical in quick 
succession, noting the times by a stop-watch with as great pre- 
cision as possible, and denote the difference of the altitudes, or 
of the zenith distances, by rf{^, and the difference of the times by 
dty we shall have 

cos ip = cosec A (371) 


The observation being made near the prime vertical, an error in 
the supposed azimuth A will have but small influence upon the 
result. If the observation is exactly in the prime vertical, or 
within a few minutes of it, we may put 

10 at 

This exceedingly simple method, though not susceptible of 
great precision, may be very useful to the navigator, as it is 
available when the sun is exactly east or west, and, consequently, 
when no other method is practicable, and, moreover, requires 
no previous knowledge of the time or the approximate latitude, 
or of the star's declination.f 

Example. — 1853 July 8, Prestel observed, near the prime 
vertical, the time required by the sun to change its altitude by a 
quantity equal to its apparent diameter, by observing with a 
sextant first the contact of the lower limb with its image in an 
artificial horizon, and then the contact of the upper limb with 

♦ Pristil, in Attnm, Naeh., Vol. XXXVII. p. 281. 

f Since the star's declination is not required, this method has the additional 
adrantage (which maj at times be of great importance to the traTeller) of being 
practicable without the use of the EphemertM, This feature entitles this method to a 
prominent place in works on narigation. 


its image, the sextant reading being the same at both observi- 
tions, namely, 30° 15' 0". He found 

Contact of lower limb, 4* 43* 34-. P.M. 
" upper " 4 47 5 .6 

3 31.5 

The sun's diameter was 31' 32". Hence we have 

d: = 31' 32" = 1892" log 3.2769 

dt = 3" 31-.5 = 211-.5 ar. co. log 7.6747 

log -fV 8.8239 

^ = 53^ 23'.5 log cos ip 9.7755 

The azimuth, however, was not exactly 90°, but about 88® 20^. 
Hence we shall have, more exactly, 


A = 88<> 20' log cosec A 0.0002 

y> = 53 22.3 log cos ^ 9.7757 

It is evident that the method will be more precise in high lati- 
tudes than in low ones. 


First Method. — By Meridian Altitudes. 

200. This is the most common, as well as the simplest and 
most reliable, of the methods used by the nax-igator. The alti- 
tude is observed with the sextant (or quadrant) from the sea 
horizon, and, in addition to the corrections used on shore, the 
dip of the horizon is to be applied. The true altitude being 
deduced, the latitude is found by (277) or (278), Art. 161. 

At sea the time is seldom so well known as to enable the 
navigator to take the star at the precise instant of its meridian 
passage. But the meridian altitude of a star is distinguished as 
the greatest, to secure which the observer commences to measure 
the star's altitude some minutes before the approximately com- 
puted time of passage, and continues to obsen^e it until he per- 
ceives it to be falling. The greatest of all his measures is then 
assumed as the meridian altitude. 


The most common practice in the case of the sun is to bring 
the lower limb, reflected in the mirrors of the instrument, to 
touch the sea horizon seen directly (a few minutes before noon), 
and then by the tangent screw to follow the sun as long as it 
rises, never reversing the motion of the screw ; as soon as the 
sun begins to fall, the limb will appear "to dip" in the sea by 
lapping over the line marking the horizon. Hence, when the 
sun "dips," the observation is complete, and the instrument is 
read oflT. But, as the waves of the sea cause the ship to rise and 
fall, the depression of the sea horizon is constantly fluctuating 
by the small amount due to the change in the height of the 
observer's eye : it is, consequently, impossible to keep the sun's 
reflected image in constant contact wdth the horizon. Expe- 
rienced observers advise, therefore, to observe and read oS 
separate altitudes in rapid succession, continuing until the 
numbers read off* decidedly decrease ; the greatest is then taken 
as the meridian altitude,* or, still more accurately, the mean of 
tlie greatest and the tT\'o immediately adjacent may be taken as 
the meridian altitude, free from the mequalities produced by the 
motion of the eye. 

201. The greatest altitude, however, is not the meridian alti- 
tude, except in the case of a fixed star. To find the correction 
for a change of declination, we have, for the time (if) from noon 
when the sun is at the greatest altitude, the formula (294), or 

tf = 

A^ sin (<p — d) 

810000 sin 1" cosy^cos^ 

in which ^d is the hourly change of declination expressed in 
seconds. The reduction of the maximum altitude to the meri- 
dian altitude is the quantity y. Art. 172, or 

(15 dy sin 1" cos ^ cos^ 

2 sin (f — d) 

These formulse give t? in seconds of time and y in seconds of arc. 
For nautical use, let 

a = the change of altitude (expressed in seconds of arc) in 
one minute of time from the meridian ; 

* Rapib, Practice of Navigation (4th edition, 1852), p. 226. 
Vol. L— 20 


then, by (287), putting / = 60-, 

810000 sin 1" cos ^ cos ^ 

a = 

2 sin (f — d) 

and therefore 



""Uo/ 4a 

The value of a is given in Bowditch's Navigator, Table XXXIL, 
with the arguments <p and d. 

If, xoe express a5 in minutes of arc^ we shall have t? in minutes of 
time and y in seconds of arc, by the formulae* 

* = — y = - — — r374) 

These formulse may be used also for the moon or a planet. The 

greatest value of a5 for the sun is 1', namely, at the equinoxes 

when 5 = 0; and in this case, if the latitude is 70°, wo have 

a = 0.7 and 

y = — == 0".36 

^ 4 X 0.7 

a quantity altogether insensible in nautical practice. 

For the moon, however, we may have ^8 = 18', and for 
f = 70° the least value of a = 0.6, whence 

= ^'^^y = 135" = 2' 16" 
^ 4 X 0.6 

Even this (which, it must be observed, is for an extreme case) 
is usually neglected by navigators, who regard observations of 
tlie moon for latitude as but approximations, on account of the 
frequent indeterminate character of the sea horizon as seen 
under the moon.f 

202. When the ship is in motion, the change of latitude pro- 
duces the same effect upon the obscrv^ed maximum altitude as 
an equal change of declination. Thus, as in the last example 
of the preceding article, if a ship in latitude 70° sail due north 

♦ BowDiTCH, Practical Navigator^ p. 169. 

t Rapkb, Practice of Navigation (4th edition), pp. 177, 226, 230. 


or due south at the rate of 18 miles per hour, the maximum 
altitude will exceed the meridian altitude by 2' 15". 

Second Method. — By Reduction to the Meridian when the Time is 


203. "When the meridian observation is lost in consequence 
of clouds, circummeridian altitudes may sometimes be obtained. 
The most convenient method of reducing them at sea is that of 
BowDiTCH. In his Table XXXTL he gives the value of a com- 
puted by (373); and in Table XXXIII. the value of t\ t being' 
reduced to minutes. Each observed altitude h is then reduced 
to the meridian altitude hy by the formula (287), or 

h^ = h + ai^ (376) 

and a number of altitudes are reduced at once by the same 
formula, by taking for h the mean of all the altitudes, and for t^ 
the mean of all the values of t^. If the observer has no tables, 
he can readily compute a by the formula 

a = l".9635 r,t^* = [0.2930] £S!!^^ (376) 

8m(f> — d) "" ■'8m(f> — d) ^ ^ 

Bowditch's table for f^ extends, however, only to t = 13"*. 
When the observations are more than 13"* from the meridian, 
he reduces the observation to the meridian by the formula (282), 

cos Ci = sin A 4" cos f> cos d (2 sin* } f) 

employing a table of log. versed sines for the value of 2 sin* J^; 
a table of natural sines for sin h and cos f ^ ; and the table of 
logarithms of numbers for the value of the last term. I prefer 
the formula (283), 

. , ,, ,v cos cp cos ^ sin* } < 
sm 1 (A, — A) = ^ 

^ cos J (A, + A) 

which effects the reduction by a single table. 

Third Method. — By Two Altitudes near the Meridian when the Time 

is not known. 

204. As it frequently happens at sea that the local time is 
uncertain, the method I have proposed in Art. 195 will be found 

■I ■ ^1* I 


of great use to the navigator. Any two altitudes h and h' being 
observed near the meridian, r being one-half the chronometer 
interval between them, corrected for rate, expressed in minutes, 
and a being found by (376), or from Bowditch*s Table XXXIL, 
we have the meridian altitude by the formula 

which may be computed without the use of logarithms. 

Example. — The approximate latitude being 38® N., the de- 
clination at noon 1° 48' 9" S., the height of the eye above the 
sea 19 feet, suppose the following observations taken : 



3" — 8» 0-22'.5 

h' — 50« 



T — 8 10 13 .5 

h =50 



2) 9 51 

h — h' = 



T — 4 55.5 



T« — 24.2 

}(A + A') — 50 



a = 2".4 

AT* — Ist corr. — 



- h')y = 625 

\V 2d « 


Mend. alt. — 50 



Dip — 



Semidiamctcr — + 



Eefr. and par. = — 


A, — 50 



C, -39 


53 N. 

i, 1 


9 S. 

SP = 37 48 44 N. 

Tlie accuracy of the result depends in a great degree upon 
the accuracy with which the difference of altitude is obtained. 
If in t!ie above example this difterence had been 2' 40", or 1' 
too great, we should have found J(A — A') = 40", and the 2d 
correction — ajJ*^- -- 28" : consequently the resulting latitude 
would have been only 17" too small. Since the same causes of 
error, such as displacement of the sea horizon by extraonlinary 
refraction, unknown instrumental errors, &c., aftect both altitudes 
alike, the difference will usually be obtained, even at sea, within 
a quantity much less than V. The most favorable case is that 


in which the altitudes are equal and the 2d correction, conse- 
quently, zero. It will be well, therefore, always to endeavor to 
obtain altitudes on opposite sides of the meridian. 

We may also obtain an approximate value of the time from 
the same observation ; for we have for the hour angle of the 
least altitude h\ Art 195, 

a = ^^^-^^ + 1(2"- T) 


Thus, in the above example we have 

i(A-AO ^ 25 ^ 2«1 
ar 2.4 X 4.9 

1(7'— r) = + 4 .9 

f = + 7 .0 

The apparent time of the observation of the least altitude was, 
therefore, 0* 7*. 

Fourth, Method. — By Three. Altitudes near the Meridian when the 

Time is not known. 

205. The method of Art. 196 does not require even the rate 
of the chronometer to be known ; but it is hardly simple enough 
for a common nautical method. But a very simple method will 
be obtained if we take three altitudes at equal intervals of time. 
Suppose the second altitude is observed at the (unknown) time 
Tivora the meridian passage, the first at the time T— Xy the 
third at the time T + x; then we have, by (363), 

\=h -j-a(T—xy 

\=h'' + a(T+xy 

Subtracting the half sum of the first and third equations from 
the second, we deduce 


The difference of the first and third gives 


which substituted in the second equation gives Aj. 

If then we put a for cu?^ the computation is expressed by the 
following simple formulae : 

a =A'— 1(A + A") 


^ ' a 

Example. — The following three altitudes were observed at 
equal intervals of time near the meridian : 

h = 43^ 8' 20" h! = 43^ 15' 30" h" = 43^ 4' 0" 

i(A + A")=43 6 10 

a = 9 20 = 660" 

\(h — K')= 1 6= 65 

Hence the reduction of the middle altitude to the meridian is 

[\(h^r)Y ^ 65« _ g, 
a 560 

which added to h' gives 

h^ = 43*' 15' 38" 

Instead of equal inter\'al8 of time, we may employ equal inter- 
vals of azimuth (Art. 197), and still reduce the altitudes by (377); 
but this would be practicable only on land. 

Fifth Method, — By a Single Altitude at a given Time. 

206. This is the method of Art. 164, which, however, should 
be restricted, at sea, to altitudes taken not more than one hour 
from the meridian, as the time is always imperfectly known and 


the error in the latitude produced by an error in the time 
increases very rapidly as the star leaves the meridian and ap- 
proaches the prime vertical (Art. 166), and the method fails 
altogether when the star is in the prime vertical. It may, how- 
ever, sometimes be verj' important to determine the latitude, at 
least approximately, when the sun is nearly east or west; and 
then the follo>ving method may be used. 

Sixth Method. — By the change of Altitude near the Prime Vertical 

207. This is the method of Art. 199. In the morning, when 
the sun has arrived within 1° of the prime vertical as observed 
with the ship's compass, bring the image of the sun's upper 
limb, reflected by the sextant mirrors, into contact with the sea 
horizon, and note the time ; let the sextant reading remain un- 
changed, and note the time when the contact of the lower limb 
occurs. In the afternoon, begin with the lower limb. Then, 
taking the sun's semidiameter = S from the almanac, and put- 
ting the difierence of the chronometer times = r, we have 

cos sp = — = [9.1249] - (378) 

15 r ^ 

This is evidently but a rough method, only to be resorted to in 
cases of emergency. With the greatest care in observing the 
contacts, and in latitudes not less than 45°, the result cannot be 
depended upon within from five to ten minutes; but even this 
degree of accuracy may, in many cases at sea, be quite satis- 

Seventh Method, — By the Pole Star. 

208. This method, though confined in its application to north 
latitudes, is very useful at sea, as it is available at all times when 
the star is visible and the horizon sufliciently distinct, and does 
not require a more accurate knowledge of the time than is 
usually possessed on shipboard. The complete discussion of it 
has been given in Art. 176 ; but for those who wish only the 
nautical method, and have passed over that article, I add the 
following simple investigation, which is sufliciently precise for 
the purpose. 

Let ZNy Fig. 27, be the meridian ; Z the zenith of the ob- 
server ; P the pole ; AN the horizon ; S the star, which describes 



a small circle ST about the pole at the dis- 
tance PS=p; ZSA the vertical circle of the 
star at the time of the observation; SA the 
true altitude = A, deduced from the observed ; 
SPZ the star's hour angle = t ; PN the lati- 
tude = (p. 

Draw SB perpendicular to the meridian: 
then, since SP is small in the case of the pole 
star (about 1° 30'), we may regard PSB as a 
plane triangle, and hence we have 

PB = PS, COB SPB =pcost 
and, since BN differs very little from SA, 

that is,* 
K we put 

we have 
and hence 

PN= ^i\r— PB = SA — PB 
<p =zh — p cos t 

= the sidereal time, 

a = the star's right ascension, 

t= e —a 

fp = h — ;) cos (6 — o) 


If then p and a be regarded as constant, the term p cos (0 — a) 
may be given in a table with the argument 0, as in Bowditch's 
Navigator, p. 206. But the polar distance and right ascension 
of the pole star vary so rapidly that in a few years sucli a table 
affords but a rude approximation. The direct computation of 
the formula with the values of p and a obtained from the 
Ephemcris for the day of the observation is preferable. 

Example. — 1856 March 10, from an altitude of Polaris ol)- 
served from the sea horizon, the true altitude h was deduced as 
below. The time was noted by a Greenwich chronometer 
which was fast 5"* 30*. The longitude was 150° 0' W. 

* If we compare this with the more exact formula (300), we see that the error of 
the nautical method is J p^ sin 1" sin^ t tan h, which is a maximum for t = 90^. 
Taking p .= 1® 30', this maximum is 70''.7 tan ^, which amounts to 3' when p -^ 
68® 30'. 


Chronometer 19» 12- 42* h = SV W. 

Correction — 6 30 

Gr. M. T. 19 7 12 p = V 27' 18" 

Longitude * 10 = 8r.3 

Local M.T. 9 7 12 logjp 1.9410 

Sid. T. Gr. noon 23 13 23 log cos t n9.5234 

Corr. for 19^ 7* + 3 8 log;>co8f nl.4645 — j)C08f= + 29.1 

e= 8 23 43 f = 31 39.1 

a= 1 5 44 

t= 7 17 69 
= 109<'2y 45" 

Eighth Method. — By Two Altitudes with the elapsed Time between 


209. This method may be successfully applied at sea, and" is 
the most reliable of all methods, next to that of meridian or cir- 
cummeridian altitudes. The formulre fully discussed in Arts. 
178 to 183 may be directly applied when the position of the ship 
has not changed between the obsenrtitions. 

But, since there should be a considerable difference of azimuth 
between the observations, the change of the ship's position in 
the interval will generally be sufficiently great to require notice. 
All that is necessary is to apply a correction to the altitude ob- 
served at the first position of the ship, to reduce it to what it would 
have been if observed at the second position at the same instant 
To obtain this correction, let J^', Fig. 28, be 
the zenith of the observer at the first observa- 
tion, S the star at that time ; Z his zenith at 
the second observation, and S' the star at that 
time. The first observation gives the zenith 
distance Z'S^ the second the zenith distance 
ZS\ Joining the points S and S' with the 
pole P, it is evident that the hour angle SPS' 
is obtained from the observed difference of 
the times of observation precisely as if the 
observer had been at rest. We have, there- 
fore, only to find ZS in order to have all the data necessary for 
computing the latitude of Z by the general methods. 

The number of nautical miles run by the ship is the number 
of minutes in the arc ZZ' ; and, since this will always be a suffi- 


ciently small number, if we draw ZA perpendicular to SZ\ we 
may regard ZAZ' as a plane triangle, and take 

ZS = Z'S — AZ' 

ZS = Z'S — ZZ' COS ZZ'S (380) 

The angle ZZ^S is the diftercnce between the azimuth of the 
star at the first observation and the course of the ship; and this 
azimuth is obtained with sufficient accuracy by the compass.* 

Employing the zenith distance thus reduced and the other 
data as observed, the latitude computed by the general method 
will be that of the second place of observation. In the same 
manner we can reduce the second zenith distance to the place of 
the first, and then the latitude of the first place will be found. 

210. The problem of finding the latitude from two altitudes is 
most frequently applied at sea in the case where the sun is the 
observed body, the observation of the meridian altitude having 
been lost. The computation is then best carried out by the 
fomml^e (315), (316), (317), (318), employing for 3 the mean 
declination of the sun, — i.e. the declination at the middle time 
between the two obser\^ations, — and then applying to the result- 
ing latitude the correction a^ found by the formula (323). To 
save the navigator all consideration of the algebraic signs in 
computing this correction, it will be sufficient to observe the 
following rule : 1st. When the second altitude is the greater^ ^pply 
this correction to the computed latitude as a northing when the 
sun is moving towards the norths and as a southing when the sun 
is moving towards the south ; 2d. When the first altitude is the 
greater, apply the correction as a southing when the sun is moving 
towards the north, and as a northing when the sun is moving 
towards the south, 

* If wc wish a more rigorous process, we must consider the sphericftl triangle 
ZZ'Sj in which we have the observed zenith distance Z'S .^ (C). the required lenith 
distance ZS -.= C, the distance run by the ship Z'Z = </, the di£fcreDce of the star's 
azimuth and the ship's course ZZ'S' = a, and hence 

cos ^ = COS ^' COS d -\- sin ^' sin d cos a 

which deTeloped gives 

(i = C — d cos a -f J ^ sin 1" cot ^' sin* a 

the last term of which expresses the error of the formula given in the text. 


If the computer chooses to neglect this correction, he should 
employ the mean declination only when the middle time is 
nearer to noon than the time of the greater altitude. In all other 
cases he should employ the declination for the time of the 
greater altitude (Art. 183). 

211. DouwEs's method of ^^ double altitudes.''* — This is a brief 
method of computing the latitude from two altitudes of the sun, 
which, though not always accurate, is yet sufficiently so when 
the interval between the observations is not more than 1*, and 
one of them is less than 1* from the meridian. 

Let A and h' be the true altitudes, d the declination at the 
middle time, T tmd T' the chronometer times of the observa- 
tions, i and V the hour angles. The elapsed apparent time X is 
found from the times 7 and T' by (322), but it is usually suffi- 
cient to take X=^T—T. We then have V=t^rX\ and by the 
first of (14) we have 

sin A = sin ^ sin d -f- cos ip cos 5 cos t 

sin K = sin f> sin ^ + cos ^ cos d cos (t -f X) 

The difference of these equations gives 

sin h — sin A' = 2 cos ^ cos d &\n (t -\- i X) sin iX 

If we put i^ = the middle time, or 

t,= t-\-hX 
we deduce 

„ . . sin A — sin A' .^o-. 

2 sm fj, = (381) 

cos ^ cos ^ sin i A 

which gives t^ by employing the supposed latitude for ip in the 
second member. We then have 

t = % — \X 

and the meridian zenith distance fj is found from the greater 
altitude A by the formula (Art. 168) ' 

cos Ci = sin A 4- cos <p cos d (2 sin* 1 1) 

* The method of finding the latitude by two altitudes is commonly called by nayi- 
gators '*the method of double altitudes/' — an obvious misnomer, as double means 
twice the same. 


and finally the latitude by the formula y = Ci + ^- Since we 
employ an assumed approximate latitude, we shall have to repeat 
the process when the computed latitude differs much from the 

This is the form of the method as proposed by Douwes and 
adopted in Bowditch's Navigator; but the following form is still 
more simple, as it requires only the table of logarithmic sines. 
The formula for t^ may be written thus : 

cos i (A + h') siniCh — h') 
sm t^ = ^^ — ' ^ ^^ ^ 

cos ^ cos ^ siniil 
then, as before, 

and the reduction of h to the meridian altitude h^ is found by 

. , ,, , V cos <p cos ^ sin* it 

sm 1(\ -h)= 

cos i (A, + A) 

Adding h^— hto A, we have the meridian altitude, from which 
the latitude is deduced in the usual manner. If the greater 
altitude is within the limits of circummeridian altitudes, it will 
of course be reduced by (284). 

The chief objection to this method is that the computation 
must be repeated when the assumed latitude is much in error. 
It can also be shown that unless the observations are taken as 
near to the meridian as we have above supposed, the computed 
value of the latitude may in certain peculiar cases be more in 
error than the assumed value, so that successively computed 
values will more and more diverge from the truth. The methods 
referred to in the preceding articles are, therefore, generally to 
be preferred. 

212. The latitude may also be found from two altitudes by 
the simple method proposed by Captain Sumner, for which see 
Chapter Vm. 




213. The longitude of a point on the earth's surface is the 
angle at the pole included between the meridian of the point 
and some assumed ^r^^ meridian. The difterence of longitude 
of any two points is the angle included by their meridians. 
These definitions have been tacitly assumed in Art. 45, where 
we have established the general equation 

£ = 2; — T (382) 

in which (Art. 47) 7J, and T are the local times (both solar or 
both sidereal) reckoned respectively at the first meridian, and at 
that of any point of the earth's surface, and L is the west 
longitude of the point. 

As an astronomical question, the determination either of an 
absolute longitude from the first meridian, or of a difterence of 
longitude in general, resolves itself into the determination of 
the difference of the time reckoned at the two meridians at the 
same absolute instant.* The various methods of finding the 
longitude which are treated of in this chapter differ only in the 
mode by which the comparison of the times at the two meridians 
is efiected. 


214. The difference of longitude between two places A and 
B being required, let a chronometer be accurately regulated at 
A^ that is, let its correction on the time at that place and its 
daily rate be determined by the methods of Chapter V. ; then 
let the chronometer be transported to -B, and let its correction 

* The astronomical difference of longitude may differ Arom the geodetic difference 
for the same reason that the astronomical latitude differs from the geodetic, Arts. S6 
and 160. 


on the time at that place be determined at any instant. The 
time reckoned at A at this last instant is also known from the 
correction and rate first found, provided the rate has not changed 
in transportation; and hence the difference of times at the same 
absolute instant, and consequently the difference of longitude, 
are found. 

aT, dT=the correction and rate determined at A at the 
time T, by the chronometer, 
aT' = the correction determined at B at the time 
T' = T -\- tj t being the interval by the chro- 
nometer ; 

then, at the instant T+ t the true time 

at^is T +t + liT+t.dT 

'' B T -\-t + aT' 

and hence the difference of longitude is 

L = £^T -^-t.dT—i^T (383) 

Thus, the longitude is expressed as the difference of the two 
chronometer corrections at the two places; and the absolute 
indications of the chronometer do not enter, except so far as 
they may be required in determining the inter\'al with which 
the accumulated rate is computed. In this expression dT\^ the 
rate in a xniit of the chronometer (an hour, or a day, solar or sidereal), 
and T' — T must be expressed in that unit. 

Example. — At Greenwich, May 5, mean noon, a mean time 
chronometer marks 23* 49"* 42*.75, and its rate in 24 chronometer 
hours has been found to be gaining 2'.671. At Cambridge, Mass., 
May 17, mean noon, the same chronometer marks 4* 34* 47'.28; 
what is the longitude of Cambridge ? 

We have 

T= May 4, 23* 49" 42v75 a T= + 0* 10- 17-.25 ^7= — 2*.671 

T+f= " 17 , 4 34 47.28 

t= 12" 4»45- 4*. 53 = 12^.198 


A!r+f.^!r= + 0* 9-44«.67 

Ar = ~4 34 47.2 8 

i=:_^4 44 31.95 


KoTS. — It is proper to distinguish whether the given rate is the rate in a chrono- 
meter unit or in a true unit of time; although the difference will not be appreciable 
unless the rate is unusually great. If the rate is 20* in 24* by the chronometer, it will 
be 20" =i: COOd in 24* of solar time. 

215. When the chronometer is carried from point to point 
without stopping to rate it at each, it is convenient to prepare a 
table of its correction for noon of each day at the first station, 
from which the correction for the time of any observ^ation at a 
transient station may be found by simple interpolation. 

After reaching the last station, it is proper to re-determine the 
rate, which will seldom agree precisely with that found at the 
first. In the absence of any other data aftecting the rate, we 
may assume that it has changed uniformly during the whole 
time. It is convenient to compute the longitudes first upon the 
supposition of a constant rate, and then to correct them for the 
variation of rate, as follows. Let 

AT,dT= the correction and rate at the time T, found at 
the first station, 
d'T=the rate found at the last station at the time 

and put 


X = — — (384) 


then X is the increase of rate in a unit of time. If an observa- 
tion at an intermediate station is taken at the time 7^ + ^ we 
must compute the accumulated rate for the interval /, which is 
eftected by multipljdng the mean rate during this interval by the 
interval. But, upon the supposition of a uniform increase, the 
mean rate from the time T to the time T + iis the rate at the 
middle instant T -\- ^t, and this rate is dT+ \tx. Hence the 
chronometer correction on the time at the first station at the 
instant T -\- ioi the supposed observation is 

£^T-\-t{dT+ \tx^ = £^T-\-t.dT+ \Vx (385) 

A longitude assigned to an intermediate station at the time 
T + i^ by employing the original rate ST^ will therefore require 
the correction + ^t^x, observing always the algebraic signs of x 
and the longitude. 


If a number m of chronometers have been employed, and each 
determination of a longitude is the mean of the 97) values which 
thej have severally given, the longitude assigned upon the sup- 
position of constant rates is to be corrected by the quantity 

V X, + X, + X, + &c. + x^ 
2 m 

in which x^ x^ &c. are the increments of the rates of the several 
chronometers in a unit of time. If then we put 

s = the sum of all the total increments during the whole 
interval n, or the sum of the values oi d'T — ^Tfor 
the several chronometers, 


q = 


•we shall have 

Correction of a longitude at a time T + t = f.q (386) 

Example.* — ^In a voyage between La Guayra and Carthagena, 
calling on the way at Porto Cabello and Cura9oa, the following 
observations having been made, the relative longitudes are re- 

By observations at La Gua}Ta on May 22 and 28, the cor- 
rections and rates of chronometers F, 3/, and P at the mean 
epoch May 24*'.885 were as follows : 



Chron. F. 

— 4*33- 7'.80 

+ 0-.77 


— 4 17.40 

— 4.54 


— 69 43.70 

— 1.47 

On arrival at Porto Cabello, the corrections on the mean time 
at that place on June &'.870 were ascertained to be — 



— 4» 37- 15'.80 


4 5 31.28 


— 5 14 13.38 

At Cura^oa the corrections on June 12^.890 were — 

* Shadwill, Notes on the Management of Chronometert, p. 111. 




— 4» 40- 59'.20 


— 4 9 55.53 


— 5 18 8.24 

And finally, at Carthagena, observations on the 25tli and 29th 
of June gave the corrections and rates at the mean epoch June 
27*'.0 as follows : 




— 5» 7-23'.55 

+ 0'.85 


4 37 47.98 

— 5.90 


— 5 44 34.42 

+ 0.30 

Employing the rates found at La Guayra, the corrections of the 
chronometers on June 5**. 870 at Porto Cabello (for which we 
have t = ll^'.OSS), and the resulting difterence of longitude, 
are, by formula (383), are as follows : 


P. Cabello — L» QoayTa. 


— 4» 32- 58'.57 

+ 4- 17'.28 


— 41 11.81 



— 5 10 1.82 

Mean + 4 16.25 

With the same rates, we have on June 12.890 at Cura^oa (for 
t == 19**. 005) the corrections and the corresponding difterence of 
longitude, as follows: 


Ciink9oa — Lft Ooayra. 


— 4» 32- 53M7 

+ 8- 6'.03 


— 4 1 43.68 

8 11.86 


— 4 10 11.64 

7 51.60 
Mean + 8 3 .16 

With the same rates, we have on June 27** at Carthagena (for 
i = 8S*.115) the corrections and the corresponding difterence of 
longitude, as follows: 


dkT+ t.iT Cutbagena— La OaayTa. 
— 4» 32- 42'.30 + 34- 41'.25 


4 2 47.74 35 0.24 


— 5 10 32.38 34 2.04 

L— 21 

Mean + 84 34 .51 


Now, to correct these results for the changes in the rates of 
the chronometers, we have, in the interval n = 83.115, 


F. + 0*.08 

M. — 1 .36 

P. +1 .77 

and, consequently. 

5 = + .49 

^ -1-^'^^ ^ ^ 0'.002466 

^ 2 X 3 X 33.115 ^ 

Applying the correction t\ to the several results, the true 
differences of longitude from Ea Guayra are found as follows: 

Approx. diff. long. t^.q Corrected diff. long. 

P. Cabello + 4- 16-.25 + 0*.35 + 4- 16'.60 

Cura^oa +8 3 .16 + .89 +8 4 .05 

Carthagena + 34 34 .51 + 2 .70 + 34 37 .21 

But it is usually preferable to carry out the result by each 
chronometer separately, in order to judge of the weight to be 
attached to the final mean by the agreement of the several indi- 
vidual values. For this purpose we have here, by the formula 
(384), for n = 33.115, 

F, + 0.00121 
M, — 0.02054 
P. + 0.02673 

and hence the correction ^t^.x is, for the several cases, as follows : 

p. Cabello. Cura9oa. Carthagena. 

F. + 0M7 + 0-.44 + 1-.32 

M, —2.95 —7.41 —22.52 

P. + 3 .84 +9 .65 + 29 .31 

Applying these corrections severally to the above approximate 
results, we have, for the differences of longitude from La Guayra, 

V. Cabello. 




+ 4- 17'.40 

+ 8- 6'.47 

+ 34- 42'.57 









Means + 4 16 .61 +84 .05 + 34 37 .21 

agreeing precisely with the corrected means found above. 


K the chronometers have been exposed to considerable 
changes of temperature, the proper correction may be intro- 
duced by the method of Art. 223. 

216. Chronometric expeditions between itoo points. — ^Where a dif- 
ference of longitude is to be determined with the greatest 
possible precision, a large number of chronometers are trans- 
ported back and forth between the extreme points. There are 
two classes of errors of chronometers which are to be eliminated: 
Ist, the accidental errors, or variations of rate which follow no . 
law, and may be either positive or negative; 2d, the constant 
errors, or variations of rate which, for any given chronometer, 
appear with the same sign and of the same amount when the 
chronometer is transported from place to place ; in other words, 
a constant acceleration, or a constant retardation, as compared 
with the rates found when the chronometer is at rest. The 
accidental erroi's are eliminated in a great degree by employing 
a large number of chronometers, the probability being that such 
errors will have different signs for different chronometers. The 
constant errors cannot be determined by comparing the rates at 
the two extreme points, since these rates are found only when 
the chronometer is at rest ; but if the chronometers are trans- 
ported in both directions, from east to west and from west to 
east, a constant error in their travelling rates will affect the differ- 
ence of longitude with opposite signs in the two journeys, and 
will disappear when the mean is taken. These considerations 
have given rise to extensive expeditions, of which probably the 
most thoroughly executed was that carried out by Struve, in 
1843, between Pulkova and Altona.* In this expedition sixty- 
eight chronometers were transported eight times from Pulkova 
to Altona and back, making sixteen voyages in all, giving the 
difference of longitude between the centre of the Pulkova Obser- 
vatory and the Altona Observatory^ 1* 21'* 32*. 527, with a probable 
error of only 0'.039. 

Chronometric expeditions between Liverpool (England) and 

♦ Expedition ehronomitrique exSeutSe par ordrt dt Sa Majesti UEmpereur Nicolas I. 
pour la dilermination de la longitude giographique relative de V ohtervatoire central de 
Ru99ie. St. Petersburg, 1844. 

For an account of the carefully executed expedition under Professor Aibt to deter- 
mine the longitude of Valentia in Ireland, see the Appendix to the Greenwich 
Observations of 1845. 


Camhriilgti (U. 8.) were instituted in the years 1849, '50, '51, ami 
"55 by the U. 9. Coast Survoy, niider the Bupcrintctnlciico of 
Professor A. D. Bache, The results of the expeditions of 1849, 
'50, and '51, diseusaed by Mr. G. P. BosD,* proved the neeefisity 
of introducing a correction for the temperature to which the 
chronometers were exposed during the voyages, and partic-oW 
attention was therefore paid to this jwint in the expetlitioii of 
1855, the details of which were arranged by Mr. W. C. Boxo. 
The results of six voyages, — three in each direction, — according 
to the clisciiseion of Mr. Q. P. Bosd,-|- were as follows : 

Voyages from Liverpool to Cnmbridgo 4' 32" 31'.92 

" " Cambridge to Liverpool 4 32 81 .75 

Mean 4 S2 31 .&4 

with a probable error of 0'.19. In this expedition fif^ chrono- 
meters were used. The greater probable error of the result, u 
compared with Strdve's, is sufficiently explained by the greater 
length of the voyages and their smaller number. 

217. The following is essentially Steuvk's method of conduct- 
ing the expeditions and discussing the results. 

Before embarking the chronometers at tlie first station {A), 
they arc carefully compared with & standard clock the correction 
of which on the time at that station has been obtained with 
the greatest precision by transits of well-determined stare. (See 
Vol, n., " Transit Instrument.") Upon tlieir arrival at the aecond 
station (^, they are compared with the standard clock at that 
station.! From these two comparisons the chronometer correc- 
tions at the two stations become known, and, if ihe rates ant 
known, a value of the longitude is found by each chrononivtor 
by (383). But here it is to be observed tliat the rate of a chro- 
nometer is rarely the same when in motion as when at re«L It 
is necessary, therefore, to find its travcUinff rate (or wa raU, as it 
is called when the chronometer is transported by sea). Thi* 
might be efl'ecfed by finding— ^rst, the correction of the chrono- 

* Report of ibc Superintendent of Ihe U. S. CoibI Siirre; for I8A4, Appendix N*. 43. 

f Repari of Lbe SupKrintendcni of Uie U. S. Cowl Survcj for 1^00, p. 182. 

t Fur thp method of comparing ehronomclen ftnd clucki with tlie x'VWcM p««- 

cUion, ate Vol. U, 


meter at the station A immediately before starting ; secondly^ its 
correction at B immediately upon its arrival there ; and thirdly, 
having, without any delay at By returned directly to A, finding 
again its correction there immediately upon arriving. The dif- 
ference between the two corrections at A is the whole travelling 
rate during the elapsed time, and this rate would be used in 
making the comparison with the correction obtained at -B, and 
in deducing the longitude by (383). 

But, since the chronometer cannot generally be immediately 
returned from -B, its correction for that station should be found 
both upon its arrival there and again just before leaving, and 
the travelling rate inferred only from the time the instrument is 
in motion. For this purpose, let us suppose that we have found 

at the times f, if, <", f", 

the chron. corrections a, 6, ft', a\ 

the correction a at the station A before leaving ; b upon arriving 
at B; b' before leaving B; and a' upon the return to A. The 
times /, t\ V'y V"j being all reckoned at the same meridian, if we 
now put 

m = the moan travelling rate of the chronometer in a unit 

of time, 
k = the loDgitude of B west of A, 

we shall have, upon the supposition that the mean travellini^ 
rate is the same for both the east and west voyages, 

X = a +m(f ^t) — b 
>l = a'— m(r— r) — 6' 

From these two equations the two unknown quantities m and k 
become known. Putting 

r=zf—t T"=r— r 

we find, first, 

m = (^'-^)-(y-^) (387) 

in which the numerator evidently expresses the whole travelling 
rate, and the denominator the whole travelling time. Then, 


(a) = a + ^^ 
we have } (388) 

A = (a) — 6 

in which (a) is the interpolated value of the chronometer correc- 
tion on the time at JL, for the same absolute instant (^ to which 
the correction 6 on the time at B corresponds. 

Example. — ^In the first two voyages of Struve's expedition 
between Pulkova and Altona in 1843, the corrections of the 
chronometer "Hauth 31" were found, by comparison with the 
standard clocks at the two stations, as below. The dates are all 
in Pulkova time, as shown by one of the chronometers em- 
ployed in the comparison : 

At Pulkova (A), t = May 19, 21*.54 a = + 0» 6- 38M0 

"Altona lB),f = « 24,22.66 b=--l 14 39.92 

" Altona (J?), r = « 26, 10 .72 6' = — 1 14 36 .77 

" Pulkova (il), r = « 81, 0.00 a'= + 7 9.M 


T = 5' 1M2 = 5'.047, a''^a = + 81-.48 
t"=4 13.28 = 4.663, 6' — 6 = + 8.15 

8P.48 — 3M5 28*.33 . ^ ^^, 
m = = = + 2-.961 

6.047 + 4.563 9.6 

a = + 0* 6* 38-.10 
mrz= 4- 14.89 

(a) = + 6 52.99 
b = ^l 14 89.92 

X = (a)—h = + l 21 82 .91 

218. In the above, the rate of the chronometer is assumed to 
be constant, and the problem is treated as one of simple inter- 
polation. But most chronometers exhibit more or less accelera- 
tion or retardation in successive voyages, and a strict interpola- 
tion requires that we should have regard to second differences. 
If we always start from the station Ay as in the above example, 
using only simple interpolation, we commit a small error, which 
always affects the longitude in the same way so long as the 
variation of the chronometer's rate preserves the same sign. 
But if we commence the next computation with the station -B, 


SO that the two chronometer corrections' at A are intermediate 
between the two at -B, then the error in the longitude will have 
a different sign, and the mean of the two values of the longitude 
will be, partially at least, freed from the influence of the acce- 
leration or retardation. To show this more clearly under an 
algebraic form, let us suppose that we have, omitting the inter* 
vals of rest at the two stations. 

at the times 

t, f, 



the chron. corrections 

a, b, 






and that 

(I = daily rate of the chronometer at the time t, 
2fi = the daily acceleration of the rate fi after the time t, 

the true values of the four corrections, observing that b and 6' 
refer to the meridian of By will be, according to the law of uni- 
formly accelerating motion, 

a = a 

b ^a + fAT + fit'—X 

If now we find the value of (a) corresponding to b (that is, for 
the time (') by simple interpolation between the values of a 
and a% we have 

« = «+(7^) 

= a + Air + /9.T(T + 0' 

from which we obtain the erroneous longitude 

X'=(^d)—b = X+fiTr^ 

Hence the error in the longitude, by simple interpolation and 
commencing with the station JL, is dX' = ^rr'. 

In the next place, if we commence at the station B^ with the 
correction 6, employing simple interpolation between 6 and 6', 
to find the correction (6) for the time /" corresponding to a', we 




and we find the erroneous longitude 

Hence the error by simple interpolation, commencing with the 
station -B, is dk'^ = — ^r'r'' ; and the error in the mean of the 
two longitudes is 

an error which disappears altogether when the intervals r and r" 
are equal. Since the voyages are of very nearly equal duration, 
it follows that by computing the longitude, as proposed by 
Struve, commencing alternately at the two stations, the final 
result will be free from the effect of any regular acceleration or 
retardation of the chronometers. 

Example. — From the " Expedition Chronom6trique" we take 
the following values for the chronometer "Ilauth 81,** being 
the combination next following after that given in the example 
of the preceding article, commencing now with the station jB, or 
Altona : 

At Altona {B\ t = May 26, 10*.72 6 = — 1* 14- 36'.77 

" Pulkova {A\ a =z " 31, .00 a = + 7 9 .68 

« Pulkova {A), t' = June 3, 5 .62 «' = + 7 19 .36 

'' Altona (J?), r= " 7, 20 .52 6'= — 1 14 0.35 


T = 4' 13*.28 = 4«'.553 6' — 6 = + 36*.42 

t"= 4 14 .90 = 4 .621 a' — a=+ 9 .78 

m = ^^'-^^ - ''- '"^ =^-'^ = + 2>.904 
4.553 + 4.621 9.174 

6 = — 1* 14- 36*.77 
mr= + 13.22 

(b) = — l 14 23.55 
a = + 7 9 .58 

; = a — (6) = + 1 21 33 .13 
The mean of this result and that of Art. 217 is ;i = 1* 21- 38'.02. 


219. Sdatwe weighi of the longitudes determined in different voyages 
by the same chronometer. — ^From the above it appears that the 
problem of finding the longitude by chronometers is one of 
interpolation. If the irregularities of the chronometer are 
regarded as accidental, the mean error of an interpolated value 
of the correction may be expressed by the formula"" 



where r and z' have the same signification as in the preceding 
article, and e is the mean (accidental) error in a unit of time. 
The weight of such an interpolated value of the correction, and, 
therefore, also the weight of a value of the longitude deduced 
from it, is inversely proportional to the square of this error, and 
may, therefore, be expressed under the form 

where A is a constant arbitrarily taken for the whole expedition, 
BO as to give p convenient values, since it is only the relative 
weights of the different voyages which are in question. 

But if the chronometer variations are no longer accidental, 
but follow some law though unknown, a special investigation 
may serve to give empirically a more suitable expression of the 
weight than the above. Thus, according to Struve's investiga- 
tions in the case of certain clocks, the weight of an interpolated 
value of the correction for these clocks could be well expressed 
by the formulaf 

But even this expression he found could not be generally applied ; 
and he finally adopted the following form for the chronometric 
expedition : 

p = ^ (389) 

in which T is the duration of an entire voyage, including the 

♦ See Vol. II., "Chronometer." 
t Expidition Chron., p. 102. 


time of rest at one of the stations, r, t'' are the travelling times 
of the voyage to and from a station, and £* is an arbitraiy 

Although this is but an empirical formula, it represents well 
the several conditions of the problem. For^Jirsiy the weight of 
a resulting longitude must decrease as the length of the voyage 
increases ; and, second, it must become greater as the difterence 
between the tvvo travelling times r, t'' decreases, since (as is 
shown in Vol. EL., " Chronometer'*) an interpolated value of a 
clock correction is probably most in error for the middle time 
between the two instants at which its corrections are given. 

220. Combination of results obtained by the same chronometer^ 

according to their weights. — ^Let >l', X", V" be the several values 

of the longitude found by the same chronometer, according to 

the method of Arts. 217 and 218 ; and p', />", />'" their 

weights by formula (389) (or any other formula which may be 
found to represent the actual condition of the voyages) ; then, 
according to the method of least squares, the most probable 
value of the longitude by this chronometer is 

L = y^- + i>-^- + J>-^- + (390) 

V' + f + /" + 

and if the diflference between this value and each particular 
value be found, putting 

n = the number of values of A, 
c =r the mean error of L, 
r = the probable error of L, 

then we shall have 


r = 0.6745 c (391) 


where [p] denotes the sum of /?', p^', &c., and \_pvv] the sum of 
p'v'v', ;/'«;"?;", &c. 

221. Combination of the results obtained by different chronometers, 
according to their weights. — The weights of the results by ditlercut 


chronometers are inversely proportional to the squares of their 
mean errors. The weight P of a longitude L will, therefore, be 
expressed generally by 



in which k is arbitrary. For simplicitj', we may assume k = 1, 
and then by the above value of e we shall have 

p _ ("-1)^ (392) 

K, then, i', L"^ U" are the values found by the several 

chronometers by (390), P', P'', P''' their weights by (392), 

the most probable final value of the longitude is 

_ rV + T-L^^-^T"^L-^-^ 

° P' + P" + P'" + ^ 

Then, putting 

i'-io=^'i X"~Xo=^'S i'"-io=^'" *c. 

iV = the number of values of X, 

E = the mean error of L^ 
B = the probable error ofIf^ 


we have 

E = J_i^2n_ B = 0.6745 -B (894) 

222. I propose to illustrate the preceding formulae by applying 
them to two chronometers of Struvb's expedition, namely, 
"Dent 1774" and "Ilauth 31." In the following table the 
longitudes found by beginning at Pulkova are marked P, those 
found by beginning at Altona are marked -4, and the numeral 
accent denotes the number of the voyage. The weights p in the 
second column are as given by Struve, who computed them by 
the formula (389), taking K= 34560 (the intervals T, r, r" being 
in hours), which is a convenient value, as it makes the weight of 
a voyage of nearly mean duration equal to unity ; namely, for 
T= 288*, t = t' = 120*. If we express T, r, r'', in days, we take 




and we shall have Struve's values o(p by the formula 





Thus, for the first voyage, we have, from the data in the example 
of Art. 217, 

T=f"—t== 11' 2*.46 = 1K103 

r = 6'.047 t"= 4'.663 

whence, by (395), 

P = 


11.103 >/(5.047 X 4.653) 

= 1.13 

The values of i' and i" are found by (890). In applying 
this formula, it is not necessary to multiply the entire longitudes 
by their weights, but only those figures which difier in the 
several values. Thus, by "Dent 1774" we have 


1* 21- 30- -^ 2'-51 X 1.10 + 2'.83 X 102 + 2'.09 X 1.14 + Ac. 

"*" 1.10 + 1.02 + 1.14 + 4c. 

1* 21- 30* -4- 2* 46 

= 1* 21- 30* + 2'.46 



































Longitudef by 


Dent 1774. 

1* 21"» 32'. 51 
81 .69 

-f 0.37 

— 0.37 

— 0.21 

— 0.77 
-h 0.81 
-f 0.88 
-f 0.48 

— 0.53 

— 0.12 
+ 0.49 

— .60 
-f 1 .31 



Z' = l»21'"32«.46 
n = 14 

[pvv^ = 3.063 
[p] = 18.91 



13 X 18.91 


= 59.04 

= ±: 0'.09 

LoDgitudet by 



1* 21"* 32«.91 

4- ©-.SO 
-\- 0.52 
-f 0.75 
-f- 0.51 

— 0.06 

— 1.06 
-f 0.09 
-f 1.55 

— 0.38 

— 0.96 
-f 0.77 

— 0.64 

— 0.88 

— 1.69 


Z"= 1* 21'" 32'.61 [/>rp] = 9.974 
n = 15 [p] = 15.69 


r" = 




Combining these two results, we have, by (393), 

i. = P 21- 32. + 0--t6 X 59 + 0..61 X 22 _ ^^ ^l- 82. 601 
* 59 + 22 

with the probable error, by (894), 

B=± 0*.067 

This agrees very nearly with the final result from the sixty-eight 

223. In the preceding method, the sea rate is inferred from 
two comparisons of the chronometer made at the same place 
before and after the voyages to and from the second place ; and 
the correction of the chronometer on the time of the first place 
at the instant when it is compared with the time of the second 
place is interpolated upon the theory that the rate has changed 
uniformly. This theory is insufficient when the temperature to 
which the chronometer is exposed is not constant during the 
two voyages, or nearly so. I shall, therefore, add the method 
of introducing the correction for temperature in cases where 
circumstances may seem to require it. 

According to the experience of M. Lieusson, the rate w of a 
chronometer at a given temperature & may be expressed by the 
formula (see Vol. IE., " Chronometer") 

m = w^ + A:(* — d^ — k't (396) 

in which &q is the temperature for which the balance is compen- 
sated, m^ the rate determined at that temperature at the epoch 
iz=Oy t being the time from this epoch for which the rate m is 
required, k the constant coefficient of temperature, and k' that 
of acceleration of the chronometer resulting from thickening of 
the oil or other gradual changes which are supposed to be pro- 
portional to the time. 

It is evident that, since every change of temperature produces 
an increase of tw, the term A:(i? — ^^y will not disappear even when 
the mean value of & is the same as i?^. It is necessary, therefore, 
to determine the sum of the effects of all the changes. Let us, 
therefore, determine the accumulated rate for a given period of 
time r. Let tWq be the rate at the middle of this period, in which 
case we have in the formula / = 0. A strict theory requires that 


we should know the temperature at every instant ; but, in default 
of this, let us assume that the period r is divided into sufficiently 
small intervals, and that the temperature is obsen^ed in each. 
Let us suppose n equal intervals whose sum is r, and denote the 
observed values of ^ by tf <»>, m, tf (»> tf<->. The rate 

in the Ist interval is [m^ + k (^<" — #.)«] X — 
" 2d " K + A: (*<«>- d,)*] X^ 

in the nth interval is [m^ + k (^^ — 6^)^ X — 

and the accumulated rate in the time r is the sum of these 

where I^{9 — &qY denotes the sum of the n values of {& — &^. 
To make this expression exact, we should have an infinite number 

of infinitesimal intervals, or we must put - = rfr, and substitute 

the integral sign J for the summation symbol 2*: thus, the exact 
expression for the whole rate in the time r is 

m,r + kfJ(^^-{^,ydT (397) 

This integral cannot be found in general terms, since & cannot 
be expressed as a function of r ; but we can obtain an approxi- 
mate expression for it, as follows. Let ^^ be the mean of all the 
observed values of ^ ; then we have 

-n (* - *•)'= -n [(^I - ^) + (^ - ^l)]' 

in which ^^ — ^^ is constant, and, therefore, for n values we have 
-T^ («?i— ^'o)*— ^ (''i" ^o)'* Moreover, since tf^ is the mean of all 
the values of 1?, we have 2*^ (<? — <?,) = 0, and, consequently, also 
-r/2((?j-«?,)((>-t?i) = 2(«?i-t?,) -r.(i?-t?J = 0; and the'above 
expression becomes 


bence, also, 

or, for an infinite value of w, 

Thus, the required integral depends upon the integral T (^ — ^i)* dvj 

which may be approximately found from the observed values of 
i^ by the theory of least squares. For, if we treat the values of 
i^ — tf 1 as the errors of the observed values of t?, and denote the 
mean error (according to the received acceptation of that term 
in the method of least squares) by €, we have 

e^^^Jtupi (398) 

n — 1 

in which n is the actual number of observed values of tf . If we 
assume that a more extended series of values, or indeed an infi- 
nite series, would exhibit the same mean error (which will be 
the more nearly true the greater the number n), we assume the 
general relation 

in which N is any number. Hence, also, 

2:^(^-^^).i. = r6« 

-ar— 1 

N N 

and, making N infinite, 

S^i^-^ydr^r^ (899) 

Substituting this value, the fonnula (397) becomes 

or [m, + k {\ — d,)« + A'e«]r (400) 

from which it appears that vIq + h {9^ — ^^f + Are* is the mean rate 
in a unit of time for the intcr\'al r, m^ being the rate at the 
middle of the interval for a temperature t? = i?,,. For any subse- 
quent inten^al r', we must, according to (396), replace rw^ by 
m^ — k% i being the interval from the middle of r to the middle 
of r'. 


Now, let us suppose that the chronometer correction is obtained 
by astronomical observations at the station Ay at the times T^ 
and T^, before starting upon the voyage, and again after reaching 
the station 5, at the times T^ and T^ these times being all 
reckoned at the same meridian. Let a^, a^ a,, a^ be the observed 
corrections, and put 

so that r and r" are the shore intervals and r' the sea intervaL 
Let the adopted epoch of the rate m^ be the middle of the sea 
interval r' ; then, by (400), with the correction k't, the accumu- 
lated rates in the three intervals are 

X + a,-^a,=:lm, +A:(^/ _ d,)« + A:e'« ] r' \ (401) 

in which T?p tf/, t?/' are the mean temperatures in the intervals 
T, r', r", and €, e', e^' are found by the formula (898). These 
three equations determine the three unknown quantities m^ Jf, 
and X. If we put 

/ = ^^-^^^ - ^(^ - *o)'-*^' 


we have, from the first and third equations, 


which substituted in the second equation gives L If, however, 
we prefer to compute the approximate longitude without con- 
sidering the temperatures, and afterwards to correct for tempe> 
rature, we shall have 


These formulse apply to a voyage in either direction ; but in the 
case of a voyage from west to east they give X Avith the negative 

The term jA'(r'' — r) r' in the first equation of (402) will not 
be rigorously obtained if the temperatures are neglected ; but it 
is usually an insensible term in practice, as r" and r are made 
as nearly equal as possible, and k' is always very small. 

In combining the results of different chronometers employed 
in the same voyage, the weight of each may be assigned accord- 
ing to the regularity of the chronometer as determined from its 
observed rates from day to day.* 


224. Terrestrial Signals. — If the two stations are so near to each 
other that a signal made at either, or at an intermediate station, 
can be observed at both, the time may be noted simultaneously 
by the clocks of the two stations, and the difference of longitude 
at once inferred. The signals may be the sudden disappearance 
or reappearance of a fixed light, or flashes of gunpowder, &;c. 

If the places are remote, they may be connected by interme- 
diate signals. For example : suppose four stations, Ay B, (7, -D, 
chosen from east to west, the first and last being the principal 
stations whose difference of longitude is required. At the in- 
termediate stations B, C let observers be stationed with good 
chronometers whose rates are known. Let signals be made at 
three x>oints intermediate between A and B, B and C, Cand Z), 
respectively. The signals must, by a preconcerted arrangement, 
be made successively, and so that the observers at the interme- 
diate stations may have their attention properly directed upon 
the appearance of the signal. If, then, at the first signal the 
observers at A and B have noted the times a and b; at the 

* Besides the papers already referred to, sec the Report of the Superintendent of 
the U. S. Coast Surrey for 1857, p. 814. 
Vol. L— 22 



second signal the observers at ^ and Cthe times b' and c; at 
the third signal the observers at C and D the times c' and d; it 
is evident that the time at A when the third signal is made is 
a + (6'— i) + {c' — c\ at which instant the time at i> is d: hence 
the difference of longitude of A and D is 

>l = a + (6' — 6) + (c' — c) — d 


and so on for any number of intermediate stations. It is re- 
quired of the intermediate chronometers only that they should 
give correctly the differences 6'— 6, c'— <?, for which parpose 
only their rates must be accurately known. The daily rates are 
obtained by a comparison of the instants of the signals on suc- 
cessive days. Small errors in the rates will be eliminated by 
making the signals both from west to east and from east to 
west, and taking the mean of the results. 

The intervals given by the intermediate chronometers shouldf 
of course, be reduced to sidereal intervals, if the clocks at the 
extreme stations are regulated to sidereal time. 

Example. — ^From the Description GSomStrique de la Frasi^ 
(Puissant). On the 25th of August, 1824, signals were observed 
between Pari^ and Sirasburg as follows: 


19* 6" 20'.3 

Intermediate Stations. 


8* 49- 48'.2 
8 54 10.8 

9*16" 0'.2 
9 30 37.8 


19* 46- 51-.4 

The correction of the Paris clock on Paris sidereal time was 

— 36*.2 ; that of the Strasburg clock on Strasburg sidereal time was 

— 27'.7. The chronometers at B and Cwere regulated to mean 
time, and their daily rates were so small as not to be sensible in 
the short intervals which occurred. 

"We have 


Mean interval 
Eed. to Bid. int. = 

Sid. interval =19 3 .3 

4« 22*.6 
14 37.6 

19 0.2 

+ 3.1 


Paris clock 19* 6* 20*.3 Strasburg clock 19* 46* 51'.4 

Correction — 36 .2 Correction — 27 .7 

Paris sid. time 19 6 44 .1 Strasburg sid. time 19 46 23.7 

Sid. interval +19 3 .3 

Paris sid. time of the 'j 

last signal } ^^ ^4 47 .4 

Strasburg do. 19 46 23.7 

X= 0»21*36'.3 

In the survey of the boundary between the United States and 
Mexico, Major W. H. Emory, in 1852, employed flashes of gun- 
powder as signals in determining the dift*. of long, of Frontera 
and San Elciario."*" 

The signals may be given by the heliotrope of Gauss, by which 
an imago of the sun is reflected constantly in a given direction 
towards the distant observer. Either the sudden eclipse of the 
light, or its reappearance, may be taken as the signal; the 
eclipse is usually preferred. 

Among the methods by terrestrial signals may be included 
that in which the signal is given by means of an electro-tele- 
graphic wire connecting the two stations; but this important 
and exceedingly accurate method will be separately considered 

225. Celestial Signals. — Certain celestial phenomena which are 
visible at the same absolute instant by observers in various parts 
of the globe, may be used instead of the terrestrial signals of the 
preceding article : among these we may note — 

a. The bursting of a meteor, and the appearance or disappear- 
ance of a shooting star. — The difficulty of identifying these 
objects at remote stations prevents the extended use of this 

6. The instant of beginning or ending of an eclipse of the 
moon. — This instant, however, cannot be accurately observ^ed, 
on account of the imperfect definition of the earth's shadow. A 
rude approximation to the difference of longitude is all that can 
be expected by this method. 

e. The eclipses of Jupiter's satellites by the shadow of that 
planet. — The Greenwich times of the disappearance of each 

* Proceedings of 8th Meeting of Am. Association, p. 64. 

satellite, and of its reappearance, are accurately given i^ 
Ephemeris : so that an observer who has noted one of these 
]ilienoinenft basjonly to take the difFerenoe between this observed 
local time of Its occurrence and the Greenwich time gi\-en in the 
E]>hemoris, to have his absolute longitude. With telescopes of 
dili'crent powers, however, the instant of a satellite's disappear- 
ance must evidently vary, since the eclipse of the satellite take» 
place gradually, and the more powerful the telescope the longi-r 
will it continue to show the satellite. If the disappearance and 
reappearance are both observed M'ith the eorae telescope, the 
mean of the results obtained will be nearly free from this error. 
The first satellite is to be preferred, as its eclipses occur mor* 
frequently and also more suddenly. Observers who wish to 
deduce their difference of longitude by these eclipses should use 
telescopes of the same power, and obseiwe under the sauie 
atmospheric conditions, as nearly as possible. But in no case 
can extreme precision bo attained by this method. 

il. The occiiUations of Jupiter's satellites by the body of tlis 
planet. — The approximate Greenwich times of the disappearance 
behind the disc, and tlie reappearance of each satellite, are given 
in the Epheineris. These predicted times servo only to enable 
the observers to direct their attention to the phenomenon at the 
proper moment.- 

c. The transits of the satcUtteB over Jupiter's disc. — The ap- 
proximate Greenwich times of "ingress" and '•egress," or the 
first and last instants when the satellite appears projected on 
the planet's disc, are given in the Ephemeris, 

/. The transits of the shadows of the satellites over Jnpiter'n 
disc. — The Greenwich times of "ingress" and "egress" of the 
shadow are also approximately given in the Ephemeris. 

Among the celestial signals we may include also eclipsea of 
the sun, or occultationa of stars and planets by the moon, or, 
in general, the arrival of the moon at any given position in the 
heavens; but, in consequence of the moon's parnllax, these 
eclipses and occultationa do not occur at the same ab^lute in- 
stant for all observei-s, and, in general, the moon's appHrrnl 
position in the heavens is affected by both parallax and refrac- 
tion. The methods of emplojing tliese phenomena as ei^als^ 
therefore, involve special computations, and will be hereafter 
treated of. See the general theorj' of ecUpnes, and the method 
of lunar distances. 



226. It is evident that the clocks at tri^o stations, A and B^ 
may be compared by means of signals communicated through 
ail electro-telegraphic wire which connects the stations. Sup- 
pose at a time T by the clock at A, a signal is made which is 
perceived at B at the time T' by the clock at that station. Let 
ATand a 7^' be the clock corrections on the times at these sta- 
tions respectively (both being solar or both sidereal). Let x be 
the time required by the electric current to pass over the wire ; 
then, A being the more easterly station, we have the difference 
of longitude X by the formula 

>l = (T + aT) — (r'+ aT') + a- = A, + a: 

Since x is unknown, we must endeavor to eliminate it. For 
this purpose, let a signal be made at B at the clock time T'\ 
which is perceived at A at the clock time T'" ; then we have 

In these formulae X^ and ^ denote the approximate values of the 
difference of longitude, found by signals east-west and west-east 
respectively, when the transmission time x is disregarded; and 
the true value is 

Such is the simple and obvious application of the telegraph to 
the determination of longitudes; but the degree of accuracy 
of the result depends greatly — more than at first appears — 
upon the manner in which the signals are communicated and 

Suppose the observer at A taps upon a signal key* at an exact 
second by his clock, thereby producing an audible click of the 
armature of the electro-magnet at B, The observer at B may 
not only determine the nearest second by his clock when he 
hears this click, but may also estimate the fraction of a second; 
and it would seem that we ought in this way to be able to deter- 
mine a longitude within one-tenth of a second. But, before even 
this degree of accuracy can be secured, we have yet to eliminate, 
or reduce to a minimum, the following sources of error: 

* See VoL IT., ** Chronograph," for the detdls of the apparatus here alluded to. 


iBt. Tho persona] error of tlio observer who gives the signat" 
2d. The pci-9onal error of the observer who rcteives tho signal 

und ostimalea tho frncUon of a second bj- the ear; 
Sd. Tho small fraction of time required to complete the galvanic 

circnit after tho finger touches tho signal key; 
4th. Tho armature time, or the time reqnired hy the arinatare at 

the station where the signal is received, to move through 

tho space in whieli it plays, and to give the audible click; 
5th. Tho errors of tho BUppoeed dock corrections, which involve 

errors of observation, and errois in the right asceiieionut of 

the stars employed. 

For the means of contending successfully ■with these sources 
of error we are indebted to our Coast Burvey, which, under the 
superintendence of Prof. Bache, not only called into existence 
the chronogmphic instruments, but has given ua the tnoet effi- 
cient method of using them. The "method of star signals," »a 
it is called, was originally suggested by the distinguished Juttro- 
nomer Mr. S. C. "Walker, but its full development in the fonu 
now employed in the Coast Survey is due to Dr. B. A. Gould. 

227. Method of Star S^als.—The difference of longitude Imn 
tween the two stations is merely- the time required by a »t»r to 
pass from one meridian to the other, and this interx-al may be 
measured by means of a single clock placed at either stutioii," 
but in the main galvanic circuit extending trom one stntioa to 
the other. Two chronographs, one at each station, are also in 
the circuit, and, when the wires are suitably connected, the clock 
seconds are recorcied upon both. A good transit instranient is 
carefully mounted at each station. 

Wlien the star enters the field of the transit instrument at A 
(the eastern station), the observer, by a preconcerted signal with 
his signal key, gives notice to the assistants at both A and B, 
who at once set the chronographs in motion, and the clock then 
records its seconds upon both, Tlie instants of tlic star* tran- 
sits over the several threads of the reticule are also recorded 
upon both chrouographs by the tajia of the observer upon his 
signal key. When tlie star has passed all the threads, the ob- 

* Tbe olock ma;, intlnd. be Ml bii; pUce which la in telegrapUa Mmaadioa vU 
the (wo iMlioDi wboH differenoe at lonplude is to be IMukL 


server indicates it by another preconcerted signal, the chrono- 
graphs are stopped, and the record is suitably marked with date, 
name of the star, and place of observation, to be subsequently 
identified and read off accurately by a scale. Wlien the star 
arrives at the meridian of -B, the transit is recorded in the same 
manner upon both chronographs. 

Suitable observations having been made by each observer to 
determine the errors of his transit instrument and the rate of 
the clock, let us put 

T^ = the mean of the clock times of the eastern transit of 
the star over all the threads, as read from the chrono- 
graph at Af 

T, = the same, as read from the chronograph at B, 

7\' = the mean of the clock times of the western transit of 
the star over all the threads, as read from the chrono- 
graph at A, 

TJ =z the same as read from the chronograph at By 

6, e' = the personal equations of the observers at A and B 

t,t'= the corrections of T^ and 2\' (or of T.and T^') for 
the state of the transit instruments at A and B, or 
the respective "reductions to the meridian" (Vol. II., 
Transit Inst.), 

dT= the correction for clock rate in the interval T/ — 2\, 
X = the transmission time of the electric current between 

A and By 
X = the difference of longitude ; 

then it is easily seen that we have, from the chronographic 
records at J., 

>l = r/ + ^r+ t' + e' — a; — (2; + T + c) 
and from the chronographic records at -B, 

and the mean of these values is 

^ = [K^i'+^«0 + ^]-[Kri+2;) + r] + Jr+e'-e (404) 

which we may briefly express thus : 


in which 

X^= the approximate difference of longitade found by the 
exchange of star signals, when the personal eqaations 
of the observers are neglected. 

This equation would be final if e' — e^ or the relative personal 
equation of the observers, were known : however, if the observew 
now exchange stations and repeat the above process, we shall 
have, provided the relative personal equation is constant, 

in which ^ is the approximate difference of longitude found aa 
before ; and hence the final value is 

I have not here introduced any consideration of the armature 
time, because it affects clock signals and star signals in the same 
manner; and therefore the time read from the chronographic 
fillet or sheet is the same as if the armature acted instanta- 
neously.* It is necessarj', however, that this time should be 
constant from the first observation at the first station to the 
last observation at the second, and therefore it is important tliat 
no changes Bhould be made in the adjustments of the apparatus 
during the interval. 

As the observer has only to tap the transits of the star over 
the threads, the latter may be placed very close together. The 
reticules prepared by Mr. AV. AVirdemanx for the Coast Sur\-iy 
have generally contained twenty-five threads, in groups or '* tal- 
lies" of five, the eipiatorial intenals between the threads, of a 
group being 2'.5, and those between the grou])8 o' ; with an ad- 
ditional thread on each side at the distance of lO* for use in olv 
servations by '' eye and ear.'' Except when clouds inten-ene 
and render it necessarv to take whatever threads mav be avail- 
able, only the three middle tallies, or fifteen threads, are used. 
The use of more has been found to add less to the accuracy of a 

♦ Dr. B. A. GorLD thinks that the armnture time raries with the fltrcof^h of the 
battery and the distance (and consequent -weakness) of the signal; being thus liable 
to be confounded with the transmission time. The effect upon the difference of 
longitude wiU be inappreciable if the batteries are maintained at nearly the 


determination than is lost in consequence of the greater fatigue 
from concentrating the attention for nearly twice as long. 

A large number of stars may thus be ob8er\'ed on the same 
night ; and it will be well to record half of them by the clock 
at one station, and the other half by the clock at the other 
station, upon the general principle of varying the circumstances 
under which several determinations are made, whenever practi- 
cable, without a sacrifice of the integrity of the method. For 
this reason, abo, the transit instruments should be reversed 
during a night's work at least once, an equal number of stars 
being observed in each position, whereby the results will be 
freed from any undetermined errors of collimation and inequality 
of pivots. Before and after the exchange of the star signals, 
each observer should take at least two circumpolar stars to 
determine the instrumental constants upon which r and r' 
depend. This part of the work must be carried out with the 
greatest precision, employing only standard stars, as the errors 
of r and r' come directly into the difference of longitude. The 
right ascensions of the "signal stars" do not enter into the 
computation, and the result is, therefore, wholly free from any 
error in their tabular places : hence any of the stars of the 
larger catalogues may be used as signal stars, and it will always 
be possible to select a sufficient number which culminate at 
moderate zenith distances at both stations, (unless the difference 
of latitude is unusually great), so that instrumental errors will 
have the minimum effect. 

A single night's work, however, is not to be regarded as con- 
clusive, although a large number of stars may have been ob- 
sen-ed and the results appear verj' accordant; for experience 
shows that there are always errors which are constant, or nearly 
so, for the same night, and which do not appear to be represented 
in the corrections computed and applied. Their existence is 
proved when the mean results of different nights are compared. 
Moreover, it is necessary to interchange the observers in order 
to eliminate their personal equations. The rule of the Coast 
Survey has been that when fifty stars have been exchanged on 
not less than three nights, the observers exchange stations, and 
fifty stars are again exchanged on not less than three nights. 
The observers should also meet and determine their relative 
personal equation, if possible, before and after each series, as it 
may prove that this equation is not absolutely constant. 



Before entering upon a series of star signals, each observer 
will be provided with a list of the stars to be employed. The 
preparation of this list requires a knowledge of the approximate 
difference of longitude in order that the stars may be so selected 
that transits at the two stations may not occur simultaneously. 

Example. — For the purpose of finding the difference of longi- 
tude between the Seaton Station of the U. S. Coast Survey and 
Raleigh, a list of stars was prepared, from which I extract the 
following for illustration. The latitudes are 

Seaton Station (Washington) ^ = + 38* 53'.4 
Raleigh " (North Carolina) ^ = + 35 47 .0 

and Raleigh is assumed to be west from Washington 6* 30*. 

Seaton sidereal 





timeof Raleigk 

No. 5036 B.A.C. 


15* 9- 36* 

+ 33* 52' 

15* 16- 6* 



18 58 

37 54 

25 28 



27 2 

31 51 

33 32 



36 35 

26 46 

43 5 



45 43 

36 7 

52 13 



55 59 

23 12 

16 2 29 



16 4 9 

45 19 

10 39 



15 21 

46 40 

21 51 

The following table contains the observations made on one of 
these stars at the above-named stations by the U. S. Coast Suney 
telegraphic party in 1853, April 28, under the direction of Dr. 
B. A. Gould. 

In this table " Lamp W.*' expresses the position of the rotation 
axes of the transit instruments. The 1st column contains the sym- 
bols bv which the fifteen threads of the three middle tallies were 
denoted; the 2d column, the times of tnmsit of the star over 
each thread at Seaton, as read from the chronographs at Seaton; 
the 3d column, the times of these transits as read from the chro- 
nographs at Raleigh ; the 4th column, the mean of the 2d and 3d 
columns ; the 5th column, the reduction of each thread to the 
mean of all, computed from the known equatorial inter\'al8 of 
the threads ; the 6th column, the time of the starts transit over 



liie mean of the threads, being the algebraic sum of the numbers 
in the 4th and 5th columns ; and the remaining columns, the 
Baleigh observaUons similarly recorded and reduced. 

SBATON— RALEIOU, 1863 AprU 28. 

BUr No. t!M B. A. a 

e—taaOb^ UmpW. 

Rmkiiili 01m. UdipW. 

r,+ T, 
















+ ZS'.« 


11' 00 


+ »•« 



41 ja 


i4^i« u !m 


38 J9 


19 JW 




IS .71 


ao !b8 jo .79 


6U .7(1 » .70 

iO J3 



a .80 






■ 32 



3. IS M.1S 




IC 6> M .4U 3« .»> 

+ .071 34 Ai 




~ 3 .lo' 3« .11 









11 M 



12 .76'[36 .17) 



15 Tm 3vIS 

UJO 3a .42 


1S.W 3.47 


±1S.,\ 3^8 

M ,73|.S8 .80 M ,67 

Z-J 20 SS .47 


38 >0.2S !7U 

2j ;ui 3 32 

3.08lai>sl a.08 

» jel M ,70 





The numbers in the last column for each station would be equal 
if the observations and chronographic apparatus were perfect ; 
and by carrying them out thus individually we can estimate their 
accuracy. The numbers [3.67] at Seaton and [36.17] at Raleigh 
are rejected by the application of Peirce'b Criterion (see Ap- 
pendix, Method of Least Squares), and the given means are 
found from the remaining numbers. 

The corrections of the transit instruments for this star 
(i= + 86''6'.9)were 

for the Seaton instrument, t = — 0'.028 
" " Raleigh " r" ^ — .193 

The rate of the clock was iiisensibte in the brief interval 
r/ — T. Hence, neglecting the personal equations of the ob- 
Ber\-ers, the difference of longitude is found as follows: 

J(r,'+ r;) + r'^lfl'52-36'.342 

j(r,+ r.)+r = 15 46 3.36A 

i,= 6 32.978 

In this manner seven other stars w 
night, and the results were as follows : 

■ observed on the same 





Diff. from memn 

5036 B. A. C. 

6* 33-.03 




+ 0.10 

5131 « 


— 0.08 

5192 " 


+ 0.01 

5259 " 


— O.Ol 

5322 « 


+ 0.01 

5388 " 


+ 0.03 

5463 « 


— 0.08 

Moan A, = 6 32 .99 

From the residuals i?, we deduce the mean error of a sinsrle 
determination by one star, 


and hence the mean error of the value 6"* 32*.99 is 



= ± 0-.02 

But this error will be somewhat increased by those errors of the 
instruments which are constant for the night, and not represented 
in r and r', and by the errors of the personal equations yet to bo 
applied. Moreover, a greater number of determinations should 
be compared, in order to arrive at a just evaluation of the mean 

228. VcloeiO/ of the galvanic current — Recurring to the equations 
of p. 343, we find, by taking the difforence between the values 
of / given by the chronograi)hic records at the two stations, 


If the clock is at the eastern stiition (yl), the time T^ will not 
differ from TJ, except in consequence of irregularities in the 
chronographs and errors in reading them, and therefore we 
should find z solely from the times 2\' and 2\\ or 




In like manner, if the clock is at the western station, we find x 
by the formula 

a: = 1(2; -TO 

Thus, in general, the transmission time will be deduced by com- 
paring the records of the star signals made at one station when 
the clock is at the other station. 

In the above example, the clock was at Washington, and 
hence, from the record of the transit at Raleigh, we have fourteen 
values of T^— 7i'= 2x, as follows: 

+ 0'.08 

+ 0'.08 

+ .05 

+ 00 

+ .09 

+ .23 

+ .03 

— .03 

+ .14 

+ .09 

+ .09 

+ .13 

+ .10 + .00 

That these are not merely accidental residuals is shown by 
the permanence of sign, with the single exception in the case 
of the eleventh observation. The discrepancies between them 
indicate accidental variations in the chronographs, combined with 
errors in reading off the record. Taking the mean, as elimi- 
nating to a certain extent these errors, we have 

2x = 0*.077 X =. 0*.0385 

From this value of x and the distance of the stations we can 
deduce the velocity per second of the galvanic current. In the 
present instance, the length of the wdre was very nearly 300 
miles, and, if the above single observation could be depended 

upon, we should have, velocity per second = -^tt^^ttj^ = 7792 miles, 

which is doubtless too small. 

The velocity thus found, however, appears to depend upon 
the intensity of the current,* as has been shown by varying the 
battery power on different nights. It has also been found that 
the velocities dctennined from signals made at the east and west 
stations differed, and that this difference was api)arently depend- 

* It depends also upon the sectional area, molecular structure, and, of course, 
materUl, of the wire. 


eiit upon the strength of the batteries ; the velocities from signak 
east-west and signals west-east coming out more and more 
nearly equal as the strength of the batteries was increased. See 
Dr. Gould's Report on telegraphic determinations of differ- 
ences of longitude, in the Report of the Superintendent of the 
U. S. Coast Survey for 1857, Appendix No. 27. 


229. The moon*s motion in right ascension is so rapid that 
the change in this element while the moon is passing from 
one meridian to another may be used to determine the difference 
of longitude. Its right ascension at the instant of its meridian 
transit is most accurately found by means of the interval of 
sidereal time between this transit and that of a neighboring well- 
known star. For this puq^osc, therefore, the Ephemerides con- 
tain a list of moon-culminating stars, which are selected for each 
day so that at least four of them are given, the mean of whose 
declinations is nearly the same as that of the moon on that day, 
and, generally, so that two precede and two follow the moon. 
The Ephemerides also contain the right ascension of the moon*8 
bright limb for each culmination, botli upper and lower, and 
the variation of this right ascension in one hour of longitude, 
— i.e. the variation during the interval between the moon's 
transits over two meridians whose difference of longitude is one 
hour. This variation is not uniform, and its value is given for 
the instant of the passage over the meridian of the Ephemeris. 
These quantities facilitate the reduction of corresponding obser- 
vations, as will be seen below. 

230. As to the observation, let 

t^, d' = the sidereal times of the culmination of the moon's 
limb and the star, respectively, corrected for all tho 
known errors of the transit instramont, and for clock 

a, a' = the right ascensions of the moon's limb and the star 
at the instants of transit; 

then we evidently have 

a==a' + * — *' (406) 


The star and the moon being nearly m the same parallel, the 
instrumental errors which aftect i? also affect i?' by nearly the 
same quantity. We should not, however, for this reason omit 
to apply all the corrections for hwum instrumental errors, since 
by this omission we should introduce an error in the longitude 
precisely equal to the uncorrected error of the instrument. For 
if the instrumental error produces the error z in the time of the 
star's transit, the effect is the same as if the instrument were 
perfectly mounted in a meridian whose longitude west of the 
place of observation is equal to z ; but the sidereal time required 
by the moon to describe this interval z is equal to 2 + the 
increase of the moon*s right ascension in this interval. Hence 
the longitude found, by the methods hereafter given, would be 
in error by the quantity z. 

231. If the lunar tables were perfectly accurate, the true 
longitude given by the observation would be found at once by 
comparing the observed right ascension with that of the Ephe- 
meris. There are two methods of avoiding or eliminating the 
errors of the Ephemeris. In the first, which has heretofore been 
exclusively followed, the observation is compared with a corre- 
sponding one on the same day at the first meridian, or at some 
meridian the longitude of which is well established. In this 
method the increase of the right ascension in passing from one 
meridian to the other is directly observed, and the error of the 
Ephemeris on the day of observation is consequently avoided ; 
but observations at the unknown meridian are frequently ren- 
dered useless by a failure to obtain the correspondmg observa- 
tion at the first meridian. 

In the second method, proposed by Professor Peircb, the 
Ephemeris is first corrected by means of all the observations 
taken at the fixed observatories during the semi-lunation within 
which the ob8er\'ation for longitude falls. The corrected Ephe- 
meris then takes the place of the corresponding ob8er\'ation, and 
is even better than the single corresponding observation, since 
it has been corrected by means of all the observations at the 
fixed obsen^atories during the semi-lunation. 

I shall consider first the method of reducing corresponding 


232. Correspo)idmg observations at places whose difference of longi- 
tude is less than two hours. — At each place the true sidereal times 
of transit of the moon-culminating stars and of the moon*8 
bright limb are to be obtained with all possible precision : from 
these, according to the fonnula (406), will follow the right as- 
cension of the moon's limb at the instants of transit over the 
two meridians, taking in each case the mean value found from 
all the stars obscrv^ed. Put 

Xj, Xg = the approximate or assumed longitudes, 
X = the true diiferenco of longitude, 
ttj, ttj, = the observed right ascensions of the moon's bright 
limb at L^ and i, respectively, 
Ji^ = the variation of the R A. of the moon's limb for 
1* of longitude while passing from L^ to i, ; 

then we have 

;=.!^' (407) 

in which, a, — ttj and IIq being both expressed in seconds, k will 
be in hours and decimal parts. 

When the difterence of longitude is less than two hours, it 
is found to be sufficiently accurate to regard Jl^ as constant, 
provided we employ its value for the middle longitude 
Jj^ =1 h{L^ -p ij), found by interpolation from the values in the 
Ei)hemeris, having regard to second differences. 

Example. — The following]: observations were made, ^^av 15, 
1851, at Santiago, Chili, by the U. S. Astronomical Kxpeditiou 
under Lieut. Gilliss, and at Philadelphia, by Prof. Kendall: 

Object. Santiago sid. time. ' PhiUd'a ttid. time. | 

tS' Librae 15* 40- 8-.87 , 15» 45- 22v:W 

Moon II Limb ' 1() 21 30.84 : 10 21 39.11 
B. A. C. 5570 I 10 33 40 .12 | 10 32 5S M 

"We shall assume the longitudes from Greenwich to be, 

Philadelphia, L, ^ 5* 0- 39-.S5 
Santiago, 7., r_-^ 4 42 19. 

the longitude of Philadelphia being that which results from the 
last chnaiometrir expeditions of the U. S. Coast rSur\-ey, and 
that of Santiago the value which Lieut. Gilliss at first assumed. 



The apparent right ascensions of the stars on May 15, by the 
moon-cuhninating list in the Kautieal Almanac, were 

* Librae 

B. A. C. 5579 

15* 45- 22'.59 
16 32 59.20 

"We have then at Philadelphia, by (406), 

^ — ^ 

o'+i^— 1^ 

* Librae 

B. A. C. 5579 

+ 36- 16'.78 
— 11 19.85 

16* 21- 39'.37 
16 21 39.35 

and at Santiago : 

^ Librae 

B. A. C. 5579 


Mean a^ = 16 21 39 .36 

16 20 56.06 
16 20 55.92 

+ 35 33.47 
— 12 3 .28 

Mean o, = 16 20 55 .99 
I. — ttj = — 43'.37 

We shall find H^ for the mean longitude i© = J (A + -^2) 
= 4*.86, by the interpolation formula (72), or 

in which, if we put n = 

A = n = 0.405 


we have 

J = "("-^> = -0.120 

and a' and b^ are found from the values of H in the Ephemeria 
as follows : 

May 15, L. C. 142'.56 , q. 92 2d dif. 

« 15, U. C. 143 .48 ^ ' __ 0-.28 

+ 0.64 [-0.85] 

« 16, L. C. 144.12 ^^o —0.41 

'* 16, U. C. 144 .35 "^ 


H = 143'.48 a'= O'M b^=i(— 0'.28 — 0'.41) = — 0'.35 

ff^ = 143'.48 + 0'.259 + 0'.042 = 143'.781 

43 Q7 

Vol. I.— 23 


which is tlie longitude of Santiago from Philadelphia, Hence, 
if the longitude of Philadelphia is correct, we have 

Long, of Santiago == 4* 42* 33*.95 from Greenwich. 

233. Corresponding observations at places whose difference of hngi- 
tude is greater than two hours, — Having found a^ and (Z, as in the 
preceding case, we employ in this case an indirect method of 
solution. For each assumed longitude we interpolate the right 
ascension of the moon's limb from the Moon Culminations in 
the Ephemeris to fourth differences. Let 

A^y A^ = the interpolated right ascensions of the moon's 
limb for the assumed longitudes L^ and X, respect- 

If the correction of the Ephemeris on the given day is f, 
the true values of the right ascension for L^ and i, are A^ + t 
and A2 + c, the error of the Ephemeris being supposed to be 
sensibly constant for a few hours ; but their difference is 

^A, + e)-{A, + e) = A,-A, 

so that the computed difference of right ascension is the same 
as if the Ephemeris were correct. If now the observed differ- 
ence ttj — tti is the same as this computed difference, the as- 
sumed difference of longitude, or i, — Z/^, is correct ;♦ but, if 
this is not the case, put 

r=K-a,)-(^-A) (m) 


^L = the correction of the uncertain longitude, which we 
will suppose to be i, 

then Y is the change of the right ascension while the moon is 
describing the small arc of longitude lL ; and for this small 
difference we may apply the solution of the preceding article, 
so that we have at once 

aX = -^ (in hours) (409) 


^L = r X (in seconds) (409*) 

* It should be obserred, however, that one of the assumed longitudes most be 
nearly correct, for it is evident that the same difference of right ascension wiU not 
exactly correspond to the same difference of longitude if we increase or deereast 
both longitudes bj the same quantity. 


in which the value of H must be that which belongs to the 
uncertain meridian i,, or, more strictly, H must be taken for 
the mean longitude between Z^ and L^ + aX; but, as aZ/ is 
generally very small, great precision in If is here superfluous. 
However, if in any case aZ/ is large, we can first find H for the 
meridian i„ and with this value an approximate value of aX; 
then, interpolating J for the meridian i, + J aZ/, a more correct 
value of aX will be found.* 

Example. — The following observations were made May 15, 
1851, at Santiago and Greenwich : 

Object. Santiago. Qreenwich. 

^Librae 15M6-' 3'.37 15M5* 22'.37 

Moon II Limb 16 21 36.84 16 9 39.41 

B.A.C. 5579 16 83 40.12 16 32 59.17 

We assume here, as in the preceding example, for Santiago 
i, = 4* 42* 19*, and for Greenwich we have L^ = 0. The places 
of the stars being as in the preceding article, we find for 

Greenwich, a^ = 16* 9* 39'.54 

Santiago, o, = 16 20 55 .99 

tt, — ttj= 11 16.45 

The computed right ascension for Greenwich is in this case 
simply that given in the Ephemeris for May 15 ; the increase to 
the meridian 4* 42* 19*. has been found in our example of in- 
terpolation, Art 71, to be 

A^'-A^ = ll- 15'.84 
and hence 

r = + 0'.6i 

We find, moreover, for the longitude 4* 42" lO", 

H = 143'.77 

A£=-rO-.61X^^ = + 16-.28 

By these observations we have, therefore. 

Longitude of Santiago = 4* 42- 34*.28 

* This method of reducing moon culminations was developed by Walkkb, Trant* 
oetUmt of the American PhUotophieal Society^ new series, Vol. V. 


234. Reduction of moon culminations by the hourly JSphenieris. — 
The method of reduction given in the preceding aitiele is per- 
fectly exact ; but the interpolation of the moon's place to fourth 
difterences is laborious. The hourly Ephemeris, however, requires 
the use of second differences only. The sidereal time of the 
transit of the moon's centre at the meridian ij is = the observed 
right ascension of the centre = a^. K then we put 

T, = the mean Greenwich time corresponding to o, as found 

by the hourly Ephemeris, 
0j = the Greenwich sidereal time corresponding to T^, 

we have at once, if the Ephemeris is correct, 

A = e, - «i (410) 

This, indeed, was one of the earliest methods proposed, but was 
abandoned on account of the imperfection of the Ephemeris. 
The substitution of corresponding observations, however, does 
not require a departure from this simple process ; for we shall 
have in the same manner, from the observations made at 
another meridian (which may be the meridian of the Ephemeris), 

i, = e, — tta 

and hence 

; = i, - i. = (0, - e.) - (a, - a,) (411) 

and it is evident that the difference {Q^ — ©<,) of the Greenwich 
times will be connect, although the absolute right ascension of 
the Ephemeris is in error, provided the hourly motion is correct. 
The correctness of the hourly motion must be assumed in all 
methods of reducing moon culminations ; and in the present 
state of the lunar theorv there can be no error in it which can 
be sensible in the time required by the moon to pass from one 
meridian to another. 

In this method a is the rischt ascension of the moon's centre 
at the instant of the transit of the centre ; which mav be de- 
duced from the time of transit of the limb bv addino: or sub- 
tractingthc '' sidereal time of semidiameter passing the meridian,'* 
given in the table of moon culminations in the Ephemeris.* 

To find 7^1 corresponding to ttp we may proceed as in Art. 64, 

* If we wish to be altogether independent of the moon-culminating table, we can 
compute the sidereal time of semidiameter passing the meridian by the formula (we 
Vol. II., Transit Instrument), 


or as follows: Let T^ and T^-\- 1* be the two Greenwich hours 
between which a^ falls, and put 

Ao = the increase of right ascension in 1"* of mean time at 

the time T^, 
da = the increase of Aa in 1*, 
oq = the right ascension of the Ephemeris at the hour T^, 

then, by the method of interpolation by second differences, we 

12 3600 J\ 60 / 

in which the interval 7\ — T^ is supposed to be expressed in 
seconds. This gives 

^ 2 3600 

and in the second member an approximate value of 7^ may be 
used, deduced from the local time of the observation and an 
approximate longitude. A still more convenient form, which 
dispenses with finding an approximate value of 7\, is obtained 
as follows : Put 

then we have 

16(1 — A) cos d 

in which S = the moon^s semidiameter, X = the increase of the moon's right ascen- 
sion in one sidereal second, and 6 = the moon's declination, which are to be taken 
for the Greenwich time of the observation, approximately known from the local time 
and the approximate longitude. 

Or we may apply to the sidereal time (= t^j) of the transit of the limb the quantity 

15 cos 6 

and the resulting ai= -^^ dtz ^ S see 6 will be the right ascension of the moon's 
centre at the local sidereal time i^j. We then find the Greenwich time Oj corre- 
sponding to Oi as in the text, and we have 


60 (g, ~ »o) 

X = 


\ ^ 7200 A» / 

01', with sufficient accuracy, 

\ 7200*A»/ 

a: = 


Putting then 

Att 7200 Aa 

we have, very nearly, 

x = x' + a:" (413) 

As a practical rule for the computer, we may observe that x" 
will be a positive quantity when Aa is decreasing, and negative 
when Aa is increasing. 

The method of this article will be found particularly conve- 
nient when the observation is compared directly with the 
Ephemeris, the latter being corrected by the following process. 
See page 362. 

235. Peirce's method of correcting the Ephemeris,* — The accuracy 
of the longitude found by a moon culmination depends upon 
that of the observed difference of right ascension. Wlien this 
difference is obtained from two corresponding observations, if 
the probable errors of the observed right ascensions at the two 
meridians are e^ and e,, the probable error of the difference will 
be = V{e^ + fij*)- [Appendix]. But if instead of an actual ob- 
servation at Z/, we had a perfect Ephemeris, or 6,-0, the 
probable error of the observed difference would be reduced toe,; 
and if we have an Ephemeris the probable error of which is less 
than that of an observation, the error of the obser^'ed difference 
is reduced. At the same time, we shall gain the additional 
advantage that every observation taken at the meridian whose 
longitude is required will become available, even when no corre- 
sponding obscn-ation has been taken on the same day; and 

* Report of the Superintendent of the U. S. Coast Surrey for 1864, Appeadii, 
p. 116» 


experience has shown that, when we depend on corresponding 
observations alone, about one-third of the observations are 

The defects of the lunar theory, according to Peirce, are 
involved in several terms which for each lunation may be 
principally combined into two, of which one is constant and the 
other has a period of about half a lunation, and he finds that 
for all practical purposes we may put the correction of the 
Ephemeris for each semi-lunation under the form 

jr=A + Bt+ O" (414) 

in which -4, J5, and G are constants to be determined from the 
observations made at the principal observatories during the 
semi-lunation, and t denotes the time reckoned from any assumed 
epoch, which it will be convenient to take near the mean of the 
observations. The value of t is expressed in days ; and small 
fractions of a day may be neglected. Let 

a^y Oji, Og, &c. = the right ascension observed at any observa- 
tory at the dates f^ ^,, t^, &c., from the assumed 

a/,tt/,tt,',&c. == the right ascension at the same instant found 

from the Ephemeris, 

and put 

then rij, n^ n,, &c. are the corrections which (according to the 
observations) the Ephemeris requires on the given dates, and 
hence we have the equations of condition 

A + Bt^+ a* — Mj = 

A + Bt,+ a,'^n, = 

In order to eliminate constant errors peculiar to any observa- 
tory, when the observation is not made at Greenwich, the ob- 
served right ascension is to be increased by the average excess 
for the year (determined by simultaneous obscn^ations) of the 
right ascensions of the moon's limb made at Greenwich above 
those made at the actual place of observation. 



If now we put 

m = 

N = 

the number of observations = the number of equations 

of condition, 

the algebraic sum of the values of ty 

the sum of the squares of t^ 

the algebraic sum of the third powers of t^ 

the sum of the fourth powers of /, 

the algebraic sum of the values of n, 

the algebraic sum of the products of n multiplied by t^ 

the algebraic sum of the products of n multiplied by V^ 

the normal equations, according to the method of least squares, 

will be 

mA+ TB + T^C — J\r = ^ 

TA+ T^B+ T,C—N,= I (415) 

T^A+ T^B+ T^C—N^=0 J 

The solution of these equations by the method of successive 
substitution, according to the forms given in the Appendix, may 
be expressed as follows : 

T.' = 









r;' — 


( T^y 


rp It 






N! = iV, - 

n; = n,- 


n;'= nj— 






A = 

N— T^C — TB 



Then, to find the mean error of the corrected Ephemeris, we 
observe that tliis en'or is simply that of the function .1', which is 
to be found by the method of the Appendix, acconlin^ to which 
we first find the coefficients k^y Aj, /*, by the following formulie: 

vik^ = 1 

mt + r; A-, + t;' k. 

= t* 

and then, puttmg 

^^ - \/W m + A-,« t; + V ^4") 



we have 

(cX) = Me 


in which e denotes the mean error of a single observation and 
(bJC) the mean error of the corrected Ephemeris ; or, if e denotes 
the probable error of an observation, (eJC) denotes the probable 
error of the corrected Ephemeris. (Appendix.) 
If the values of A^, ij, and k^ are substituted in JSf, we shall have 

It will generally happen, where a suflBcient number of observa- 

tions are combined, that -^ is a small fraction which may be 

neglected without sensibly affecting the estimation of a probable 
error, and we may then take 



According to Peirce, the probable error of a standard observa- 
tion of the moon's transit is 0M04 (found from the discussion of 
a large number of Greenwich, Cambridge, Edinburgh, and Wash- 
ington observations) ; so that the probable error of the corrected 
Ephemeris will be equal to 31. (0'.104). 

Example. — At the Washington Observatory, the following 
right ascensions of the moon were obtained from the transits over 
twenty-five threads, observed with the electro-chronograph : 

Approx. Green. Mean Time. 

R. A. of 
3) II Limb. 

Sid. time semid. 
passing merid. 

R. A. of 3) centre 

1859, Aug. 16, 19* 

0» 8*53'.40 


0* 7-51*.34 

« 17, 20 

54 33.57 


53 30.03 

" 18, 21 

1 42 48.53 


1 41 42.76 

The sidereal time of the semidiameter passing the meridian is 
liere taken from the British Almanac, as we propose to reduce the 
observations by means of the Greenwich observ^ations which are 
reduced by this almanac. We thus avoid any error in the semi- 

During the semi-lunation from Aug. 13 to Aug. 27, the 
Greenwich observ^ations, also made with the electro-chronograph, 



gave the following corrections (= n) of the Nautical Almanac 
right ascensions of the moon : 

Approx. Greenwich Mean Time. 



1859. Aag. 

14, 13» 


— 8. 

15, 14 

— 0.26 

— 1.9 

16, 14 

— 0.49 

— 0.9 

18, 16 

— 0.63 

+ 1.2 

19, 17 


+ 2.2 

20, 17 

— 1.08 

+ 3.2 

Let us employ these observations to determine by Peirce'a 
method the most probable correction of the Epliemeris on the 
dates of the Washington observations. Adopting as the epoch 
Aug. 17th 12* or 17*'.5, the values of i are approximately as above 
given. The correction of the Ephemeris being sensibly constant 
for at least one hour, these values are sufficiently exact. We 
find then 

r— 0.8 

T, — 29.94 

T, = 10.556 

7; = 


r;— 29.83 

r,' — 6.564 



Tl'— 74.200 

»» = 6 

N — — 3'.89 

JV, — — 4'.41 


— 21'.85 

iV;'= — 3.89 


— 2.44 

J\r,"— — 1'.58 

and hence, by (416), 

C — 


B — — 0: 


A = 

— 0'.525 

The correction of the Ephemeris for any given date i^ reckoning 
from Aug. 17.5, is, therefore, 

X= — 0'.525 — 0M257f — 0*.02135f» 

Consequently, for the dates of the Washington ob8er^•ation!», 
the correction and the probable error (3/£) of the correction, 
found by (418) or (418*), are as follows: 

Aug. 16, 19* i = — 0.7 

17, 20 t = -\- 0.3 

18, 21 t = -\- 1.4 

X=-^ 0-.45 
X= — 0.56 
jr= — 0.74 

JlTc = O-.OS 
Mt = .04 
JlTc = .04 

The longitude of the Washington Ob8e^^•ato^}• may now lie 
found by the hourly Ephemeris (after applying these correo- 
tions), by the method of Art. 234. Taking the obsen-ation of 
Aug. 16, we have 



Aug. 16, 2; = 19*, R. A. of Ephemeris 
Ha = 1.8122 da = + 0.0023 a^ 

log (a, — 0,) 

ar. CO. log Aa 
log 60 

log of 




= 35- 26*.57 
= — .80 

0* 6- 47*.56 
— .45 

6 47.11 
a^ = 7 51.34 

•1 — 0^= 1 4.23 

logx'» 6.6554 

log^a 7.3617 

ar. CO. log Aa 9.7418 

log r^Ji 1-1427 
log x" n9.9016 

x =35 25.77 

19» 35- 25'.77 

9 37 24.18 

3 13.09 

Hence, Greenwich mean time = T^-\- x = 
Sidereal time mean noon 
Correction for 19* 35- 25*.77 

Greenwich sidereal time = 

Local Bidereal time = 04 = 

Longitude =5 8 11 .70 

5 16 3 .04 
7 51.34 

The observations of the 17th and 18th being reduced in the 
same manner, the three results are 

Probable error.* 


Aug. 16, 5» 8- ll'.TO 



« 17, 12.50 



« 18, 11.10 



Mean by weights = 5 8 11.74 


236. Combination of moon culminations by weights. — When some 
of the transits either of the moon or of the comparison stars are 
incomplete, one or more of the threads being lost, such observa- 
tions should evidently have less weight than complete ones, if 
we wish to combine them strictly according to the theory of 
probabilities. Besides, other things being equal, a determina- 
tion of the longitude will have more or less weight according to 
the greater or less rapidity of the moon's motion in right ascen- 

* For the computation of the probable error and weight, see the following article. 

864 LONaiTUDE. 

If the weight of a transit either of the moon or a star were 
simply proportional to the number of observed threads, as has 
been assumed by those who have heretofore treated of this sub- 
ject,* the methods which they have given, and which are obviouB 
applications of the method of least squares, would be quite suffi- 
cient. But the subject, strictly considered, is by no means so 

Let us first consider the formula 

Oj = a' + dj — ^ 
or, rather 

in which t?; and t?' are the observed sidereal times of the transit 
of the moon and star, respectively ; a' is the tabular right ascen- 
sion of the star, and a^ is the deduced right ascension of the 
moon. The probable error of a^ is composed of the probable 
errors of ??i and of a' — t?', which belong respectively to the 
moon and the star. We may here disregard the clock errors, as 
well as the unknown instrumental errors, since tliey aiiect !>| 
and I?' in the same manner, very nearly, and are sensibly elimi- 
nated in the difterence &^ — i>'. The probable error of the 
(luantity a' — i>' is composed of the errors of a' and <?'. Tlie 
I)robable error of the tabular right ascension of the moon-culmi- 
nating stars is not only very small, but in the case of correspond- 
ing observations is wholly eliniinatod ; and even when we use 
a corrected Ephcmcris it will have but little eftect, since the ol)- 
sen'^cd right ascension of the moon at the principal ob8er\'atorie8 
always depends (or at least should depend) chiefly upon these 
stars. We may, therefore, consider the error of a' — i>' as sim- 
plv the error of (?'. We have here to deal with those errors onlv 
which do not necessarily affect i?' and ??i in the same manner, 
and of these the chief and onlv ones that need be considered 
here are — 1st, the culmination error produced by the peculiar con- 
ditions of the atmosphere at the time of the star's transit, which 
are constant, or nearly so, during the transit, but are diflferent 
for different stars and on diftercnt days; and, 2d, the accidental 
error of observation. It is only the latter which can be diminished 

* NicoLAi, in the ABtronomitche Xachrichfrn, No. 26; and S. C. WALKsm, 
tions of the Americ&n Philosophical Society, Vol. VI. p. 258. 


by increasing the number of threads. Li Vol. 11. (Transit In- 
strument) I shall show that the probable error of a single deter- 
mination of the right ascension of an equatorial star (and this 
may embrace the moon-culminating stars) at the Greenwich 
Observatory is O'.OG, whereas, if the culmination error did not 
exist it would be only 0'.03, the probable error of a single 
thread being = O'.OS, and the number of threads = 7. Hence, 

c = the probable culmination error for a star^ 

we deduce* 

c = l/(0.06)»--(0.03)» = 0'.052 
If, then, we put 

c = the probable accidental error of the transit of a star over 

a single thread, 
n = the number of threads on which the star is observed, 

the probable error of i>', and, consequently, also of a' — t?', is 

^ n 

and the weight of a' — ^' for each star may be found by the 

P= — 7^ 

in which JE is the probable error of an observation of the weight 
unity, which is, of course, arbitrary. K we make p = 1 when 
w = 7, we have E = 0'.06. Substituting this value, and also 
c = 0'.052, £ = O'.OS, the formula may be reduced to the fol- 
lowing : 

P = -^8 (419) 

100 + — 

The value of a^ is to be deduced by adding to ??i the mean 

* The value of c thus found inyolves other errors besides the culmination error 
proper, such as unknown irregularities of the clock and transit instrument, &o. 
These cannot readily be separated Arom c, nor is it necessary for our present purpose. 


according to weights of all the values of a^ — tfj given by the 
several stars, or 

where the rectangular brackets are employed to express the sum 

of all the quantities of the same form. The probable error of 

the last term will be 

_ E _ 0v06 

"■ VlPl ~ ViP\ 
K now we put 

e^ = the probable error of oj, 
Cj = the culmination error for the moon, 
he = the probable accidental error of the transit of the 
moon's limb over a single thread, 
n^ = the number of threads on which the moon is observed. 

the probable error of t?i will be = ^/<?i*+ ^ \ and hence 

i l>] 

To determine c^ I shall employ the values of the other quantities 
in this equation which have been found from the Greenwich 
observations. Professor Peirce gives t^ = 0'.104, and in the 
cases which I examined I found the mean value k = 1.3. As- 
suming [p] = 4 as the average number of stars upon which Oj 
depends in the Greenwich series, we have 

(0.104). = c,.+ (5:m'+(i:p! 


c, = O'.OOl 

and the formula for the probable error of a^ observed at the 
meridian L^ is 

,.= („.09,,+ (!^-+C^' (422) 

In the case of corresponding observations at a second meridian 
Xj, the probable error e, is also to be found by this formula, and 
then the probable error of the deduced difference of right ascen- 
sion will be 


and the probable error of the deduced longitude will be 

= Ai/e,«+c,« (423) 

where, If being the increase of the moon's right ascension in 1* 
of longitude, we have 

r 3600 

^ = -jy- (424) 

But if the observation at the meridian ij is compared with a 
corrected Ephemeris (Art. 235) the probable error of which is 
J!f (0*.104), the probable error of the deduced longitude will be 

= h y^fj« + Jf « (0.104)« (425) 

Finally, all the different values of the longitude will be com- 
bined by giving them weights reciprocally proportional to the 
squares of their probable errors. 

The preponderating influence of the constant error represented 
by the first term of (422) is such that a very precise evaluation 
of the other terms is quite unimportant. It is also evident that 
we shall add very little to the accuracy of an observation by 
increasing the number of threads of the reticule beyond five or 
seven. For example, suppose, as in the "Washington observations 
used in Art. 235, that twenty-five threads are taken, and that 
four stars are compared with the moon ; we have for each star, 
by (419), 

and hence 

= ^[(„.0„,+ 2f)-H.M.1 = 0..0,, 

whereas for seven threads we have t^ = 0'.104, and therefore 
the increase of the number of threads has not diminished the 
probable error by so much as 0*.01. 

For the observations of 1859 August 16, 17, 18, Art. 235, the 
values of A are respectively 

82.1 30.8 and 28.8 

and, taking Mt = Jff (0'.104) as given in that article, namely, 

O-.Od 0*.04 and 0'.04 


with the value of e^ = 0*.097 above found, we deduce the proba- 
ble errors of the tlu'ee values of the lougitude, by (425), 

8*.5 3*.l and 2'.9 

The reciprocals of the squares of these errors are very nearly in 
the proportion of the numbers 1, 1.3, 1.5, which were used as 
the weights in combining the three values. 

237. The advantage of employing a corrected Ephemeris 
instead of corresponding observations can now be determined 
by the above equations. If the observations are all standard 
observations (represented by n^=l and [p] = 4), we shall have 
€i= €3= 0'.104, and the probable error of the longitude will be 

by corresponding observations = ht^ y/2 

by the corrected Ephemeris = he^ y/\ -\- M^ 

The latter will, therefore, be preferable when JIf < 1, which will 
always be the case except when very few observations have been 
taken at the principal observatories. 

But experience has shown that when we depend wholly on 
corresponding observations we lose about one-third of the 
observation.s, and, consequently, the probable error of the final 
longitude from a series of observations is greater than it would 
be were all available in the ratio of j 3 : v 2. Hence the proba- 
ble errors of the final results obtained by corresponding obser>'a- 
tions exclusively, and by employing the corrected Ephemeris by 
which all the observations are rendered available, are in the 

ratio I 3 : ] 1 + J/^, and, the average value of M being about 
O.G, this is as 1 : 0.G7. 

If, however, on the date of any given obsen-ation at the meri- 
dian to be determined, we can find corresponding obserx'ations 
at two princii)al obsen-atories, the probable error of the longitude 
found by comparing their mean with the given obser\'ation will 

be only ht^ ] 1.5, which is so little greater than the avenigo ern>r 
in the use of the corrected Ephemeris, that it will hardly be 
worth while to incur the labor attending the latter. If there 
should l)e three corresponding obsen'ations, the error will be 

reduced to Asj] 1.33, and, therefore, less than the average error 
of the corrected Ephemeris. 


The advantage of the new method will, therefore, be felt 
chiefly in cases where either no corresponding observation, or 
but one, has been taken at any of the principal observatories. 

238. The mean value of h is about = 27, and therefore a 
probable error of 0*.l in the observed right ascension, supposing 
the Ephemeris perfect, will produce a mean probable error of 2'.7 
in the longitude. K the probable error diminished without 
limit in proportion to the square root of the number of observa- 
tions, as is assumed in the theory of least squares, we should 
only have to accumulate observations to obtain a result of any 
given degree of accuracy. But all experience proves the fallacy 
of this law when it is extended to minute errors which must 
wholly escape the most delicate observation. The remarks of 
Professor Peirce on this point, in the report above cited, are of 
the highest importance. He says : " If the law of error embodied 
in the method of least squares were the sole law to which 
human error is subject, it would happen that by a sufficient 
accumulation of observations any imagined degree of accuracy 
would be attainable in the determination of a constant ; and the 
evanescent influence of minute increments of error would have 
the effect of exalting man's power of exact observation to an 
unlimited extent. I believe that the careful examination of 
observations reveals another law of error, which is involved in 
the popular statement that ' man cannot measure what he cannot 
see.' The small errors which arc beyond the limits of human 
perception are not distributed according to the mode recognized 
by the method of least squares, but either with the uniformity 
which is the ordinary characteristic of matters of chance, or more 
frequently in some arbitrary form dependent upon individual 
peculiarities,— such, for instance, as an habitual inclination to the 
use of certain numbers. On this account, it is in vain to attempt 
the comparison of the distribution of errors with the law of least 
squares to too great a degree of minuteness ; and on this account, 
there is in every species of observation an ultimate limit of accuracy 
beyond which no mass of accumulated observations can ever penetrate. 
A wise observer, when he perceives that he is approaching this 
limit, will apply his powers to improving the methods, rather 
than to increasing the number of observ^ations. This principle 
will thus serve to stimulate, and not to paralyze, eflbrt ; and its 

Vol. L— 24 

vivifjing influence will prevent science from stagnating into 
mere mechanical drudgery. 

" In approaching the ultimate limit of accuracy, the probable 
error ceases to diminish proportionably to the increase of the 
number of observations, eo that the accuracy of the mean of 
several determinations does not surpass that of the single detCT- 
minatious as much as it should do in conformity with the law of 
least squares ; thus it appears that the probable error of the 
mean of the determinations of the longitude of the Harrard 
Observatory, deduced from the moon-culminating observation* 
of 1845, 1846, and 1847, is 1'.28 instead of being I'.OO, to which 
it should have been reduced conformably to the accuracy of ihe 
separate determinations of those years. 

" One of the fundamental principles of the doctrine of probft- 
bilities is, tliat the probability of an hypothesis is proi>ortionate 
to its agreement with observation. But any supjtosed computed 
lunar epoch may be changed by several hundredths of a second 
without perceptibly attectiug the comparison with observation, 
provided the comparison is restricted within its legitimate limits 
of tenths of a second. Ob8er\'ation, therefore, gives no informa- 
tion which is opposed to such a change." 

The ultimate limit of accuracy in the deteimination of a 
longitude by moon culminations, according to the same distin- 
guished authority, is not less than one second of ii»u. This limit 
can probably be reached by the observations of t^vo or thre« 
years, if all the possible ones are taken ; and a longer continaance 
of them would bo a waste of time and labor. 

From these considerations it follows that the method of moon 
culminations, when the transits of the limb are employed, cannot 
come into competition with the methods by chronometers and 
occultations where the latter are practicable* 

* Id oonsequencfl of Ihe uncertainl; atlcnding Iho obMrvtlion et lb* uaa^ of 
the mgoD'a limb, it Lbs bceo prcipoaed by MaRdLIK [Aitron. AarA. No. SST) to mb- 
Btituto llio tmnait of a irell-deflDpJ luoar «pot. Tba anlj ftdempi lo carry eat ibU 
(uggeslion, I ihink, is that of ihe U, S, Coael 8QrT«y. a Kpgri upon vliieb by Mr. 
I'BTEn* Hill be round in tbe RefioTt of itie Superiatendent for I86n. p. 108. Tba 
Tsfjiiilg character of a epol as seen in Ip1csoop«s of different powen priaantt. It 
leems lo me, a *ery formidable obBtule Ut Uie tuccestful appUcation of lUf 



289. The travelling observer, pressed for time, will not unfre^ 
quently find it expedient to mount his transit instniment in the 
vertical circle of a circumpolar star, without waiting for the meri- 
dian passage of such a etar. The methods of detennining the 
local time and the instrumental constants in this case are given 
in Vol. n. lie may then also observe the transit of the moon 
and a neighboring star, and hence deduce the right ascension of 
the moon, which may be used for determining his longitude 
precisely as the culminations are used in Art. 284. 

240. But if the local time is previously determined, we may 
dispense with all observations except those of the moon and the 
neighboring star, and then we can repeat the observation several 
times on the same night by setting the instrument successively 
in different azimuths on each side of the meridian. It will not 
be advisable to extend the observations to azimuths of more than 
15° on either side. 

The altitude and azimuth instrument is peculiarly adapted for 
such observations, as its horizontal circle enables us to set it at 
any assumed azimuth when the direction of the meridian is 
approximately known. The zenith telescope will also answer 
the same purpose. But as the horizontal circle reading is not 
required further than for setting the instrument, it is not indis- 
pensable, and therefore the ordinary portable transit instrument 
may be employed, though it will not be so easy to identify the 
comparison star. 

The comparison star should be one of the well-determined 
moon-culminating stars, as nearly as possible in the same 
parallel with the moon, and not far distant in right ascension, 
either preceding or following. 

The chronometer correction and rate must be determined, with 
all possible precision, by observations either before or after the 
moon obsen^ations, or both. An approximate value of the cor- 
rection should be known before commencing the observations, 
as it will be expedient to compute the hour angles and zenith 
distances of the two objects for the several azimuths at which it 
is proposed to observe, in order to point the instrument properly 
and thus avoid observing the wrong star. 


To secure the greatest degree of accuracy, the observations 
should be conducted substantially as follows : — 

1st. The instrument being supposed to have a horizontal circle, 
let the telescope be directed to some terrestrial object, the 
azimuth of which is known (or to a circumpolar star in the meri- 
dian), and read the circle. The reading for an object in the 
meridian will then be known ; denote it by a. 

2d. The first assumed azimuth at which the transits are to be 
observed being Ay set the horizontal circle to the reading ^ + a, 
and the vertical circle to the computed zenith distance of the 
moon or tlie star (whichever precedes). This must be done a 
few minutes before the computed time of the first transit. 

3d. Observe the inclination of the horizontal axis with the 
spirit level. 

4th. Observe the transit of the first object over the several 

5th. If there is time, observe the inclination of the horizontal 

6th. Set the vertical circle for the zenith distance of the second 
object, and obser\'e its transit. 

7th. Observe the inclination of the horizontal axis with the 
spirit level. 

The instrument must not be disturbed in azimuth during these 
operations, which constitute one complete obser\'ation. 

Now set upon a new azimuth, sufticiently greater to bring the 
instrument in advance of the preceding object, and repeat the 
observation. It will often be possible to obtain in this way four 
or six observations, two or three on each side of the mendian, 
but the value of the result will not be much increased bv taking 
more than one obsen*ation on each side of the meridian. 

The collimation constant is supi)Osed to be known; but<, in 
order to eliminate any error in it, as well as inequality of pivots, 
one-half the observations should be taken in each position of 
the rotation axis. 

The azimuth of the instrument at each obser\'ation is onlv 
known from the local time, and hence the following indirei't 
method of computation will be found more convenient than the 
usual method of reducing extra-meridian transits; but the 
reader will find it easy to adapt the methods given in Vol. U. for 
such purpose to the present case. 


t,f = 

c,c' = 

^f^ = 

A,A' = 


We shall make use of the following notation : 

Tf T* = the mean of the chronometer times of transit of 
the moon's limb and the star, respectively, over 
the several threads,* 
aT, Ar'= the corresponding chronometer corrections, 

b,b'= the inclinations of the horizontal axis at the times 
T and T, 
c = the collimation constant for the mean of the 
a, a = the moon's and the star's right ascensions, 

" " declinations, 

** " hour angles, 

" " true zenith distances, 

" " parallactic angles, 

" " azimuths, 

Aa = the increase of the moon's right ascension in one 

minute of mean time, 
a5 = the increase (positive towards the north) of the 
moon's declination in one minute of mean time, 
9r = the moon's equatorial horizontal parallax, 
S = the moon's geocentric semidiameter, 
^ = the observer's latitude, 
L'= the assumed longitude, 
^L=z the required correction of this longitude, 
L = the true longitude = i' -f ^L, 

The moon's a, 5, ;r, and S are to be taken from the Ephemeris 
for the Greenwich time 3^+ a 3^+ ^'(expressed in mean time). 
The changes Aa, a5 are also to be reduced to this time. The 
right ascension and declination must be accurately interpolated, 
from the hourly Ephemeris, with second differences. 

The quantities j4, ^, y are now to be computed for the chro- 
nometer time Ty and A\ f ', q' for the time T'. Since A and A' 

* The chronometer time of passage over the mean of the threads wiU be obtained 
rigorously by reducing each thread separately to the mean of all by the general 
formula giren for the purpose in Vol. II. If, however, the same threads are 
employed for both moon and star, and e denotes the equatorial distance of the mean 
of the actually obsenred threads from the collimation axis, it will suffice (unless the 
obserrations are extended greatly beyond the limits recommended in the text) to 
take the means of the observed times at the times of passage over the fictitious 
thread the collimation of which is = c. The slight theoretical error which this 
procedure involves will be eliminated if the observations are arranged symmetrically 
with respect to the meridian. 


are required with all possible precision, logarithms of at least six 
decimal places are to be employed in their computation ; but for 
C> ?> C'j ?'> ^^^^ decimal places will suffice. The following formube 
for this purpose result from a combination of (16) and (20) : 

For the moon. For the star. 

tan M= tan (J sec ^ "j . , f tan Jf' = tan d' sec f 

^ , tan^cosJbr V ^^tli six I tan ^cos Jf 

tan -A = -r— -— ( decimals; J tan -A = -:— — - 

sm (^ — JiT) ) 'I sm (f> — M') 


tan N = cot ^ cos ^ \ / tan JV^= cot ^ cos f 

tan ^ sin iV j \ ^ . tan t sin N^ 

^°^ = TTTI^f with four ) tan gr' = 

cos(cJ + i^) decimals- ( co8(^'+^"0 

^ ^ cot (3 + ^ \ '^'''^'°'*^'' i - cot(^-+iVrO 

tan C = ^ — ' I I tan C = — ^^ — — — • 

cos q J \ cos ^ 

in which A and q are to be so taken that sin A and sin q shall 
have the same sign as sin t. 

The true azimuth of the moon's limb will be found by applying 
to the azimuth of the centre the correction 

S Fupper sign for Ist limbl 
"" sin C [lower « " 2d " J 

K we assume the parallax of the limb to be the same as that of 
the centre (which involves but an insensible error in this case), 
we next find the apparent azimuth of the limb by applying the 
correction given by (116), or 

p7:{<p — <p') sin 1" sin A' cosec C 

in which f — ^' is the reduction of the latitude, and p is the 
terrestrial radius for the latitude (p. In this expression we 
employ A^ which is tlie computed azimuth of the star, for the 
apparent azimuth of the moon's limb, since by the nature of the 
obser\'ation they are very nearly equal. 

To correct strictly for the coUimation and level of the instm- 
ment, we must have the moon's and star's apparent zenith dis- 
tances, which will be found with more than sufficient accuracy 
for the purpose by the fornmlte 

moon's app. zen. dist. = Ci = C + '^ sm C — refraction 
star's " " " =C/i=C'— refraction 


and then the reduction of the true azimuth to the instrumental 
azimuth (see Vol. IL, Altitude and Azimuth Instrument) is 

for the mooD; q= 

sin Ci tan Ci 

for the star, =p -^ qi 

sinCi' tanCi' 

the upper or lower sign being used according as the vertical 
circle is on the left or the right of the observer. The computed 
instrumental azimuths are, therefore, 

. . . , S , fiitCw — e»')8inl"sin^' c b 

(moon) ill =A ±-:— 1+ ^^ ^f qi -;— -qp 

'sinC sine sinCi tanCi 

) (427) 

(star) A,'=A'^^^qz~^ 
^ ^ sine/ tanC/ 

If now the longitude and other elements of the computation are 
correct, we shall find A^ and A/ to be equal : otherwise, put 

X = il, — il/ (428) 

then we are to find how the required correction Ai depends on x, 
supposing here that all the elements which do not involve the 
longitude are correct. Now, we have taken a and 8 from the 
Ephemeris for the Greenwich sidereal time 7" + a T + i', when 
they should be taken for the time T+ £iT+ i'+ ^L. Hence, 
if X and ^ denote the increments of the moon's right ascension 
and declination in one sidereal second, both expressed in seconds 
of arc, 

^ = ^r^ = [9.39675] Aa 
60.164 •■ -* 

we find that 

'^ = 60^ = ^'-''"^^ ^' 

a requires the correction X . ^L 
t « « — ;.aX 


and these corrections must produce the correction — xm the moon's 
azimuth. The relations between the corrections of the azimuth, 
the hour angle, and the declination, where these are so small as 
to be treated as differentials, is, by (51), 


- . C08^C08flr _ . si 

dA = dt -\ — 



sin C Bin C 

that is, 

cos d cos a , ^ . sin flr ^ _ 

Sin C sin C 

Hence, if we put 

sin C sin C 

we have 

aX = - (431) 

and hence, finally, the true longitude L'+ aX. 

241. In order to determine the relative advantages of thia 
method and that of meridian transits, let us investigate a formola 
which shall exhibit the effect of every source of error. Let 

da, 3d, dn, dS = the Corrections of the elements taken from 

the Ephemeris of the moon, 
da, dd' = the corrections of the star's place, 
dT, dT' = the corrections for error in the obs'd time^ 
d^T =z the correction of A T, 
dip = the correction of <p. 

If, when the corrected values of all the elements — ^that of the 

longitude included— are substituted in the above computation, 

A^ and -4/ become A^ + dA^ and -4/ + rf-4/, we ought to find, 


A^ + dA, = A^ + dA{ 

which compared with (428) gives 

a; = — rf^i + dA^ (432) 

We have, therefore, to find expressions for dAy^ and dA^' in 
terms of the above corrections and of aX. We have, first, by 
diflerentiating (427), 

sinC siiiC 


We neglect errors in c and 6 which are practically eliminated 
by comparing the moon with a star of nearly the same declina- 
tion, and combining observations in the reverse positions of the 


The total difierential of A is, by (51), after reducing dt to arc, 

dA = -Ibdt '\ dd — cot C sin A dtp 

sin C sin C 

consequently, also, 

.., cos^'cos^f' -.-^ , sing^ ,-, .. . .,, 

dA = : — —^ . lodf -\ — 7—^ dd' — cot C sin Adtp 

sm C sin C 

Since /^=3^4-a3^— a, we have 

dt = dT+ d^T-'da 

where rfSTand d^TmsLj be at once exchanged for 5 7" and 3^T; 
but da is composed of two parts : Ist, the correction of the 
Ephemeris, and 2d, X (j^L + dT -\- 3£kT)y which results from our 
having taken a for the uncorrected time. Hence we have, in 

lbdt = UdT+ 15 ^aT— 15^a — >l(Ai + dT+d/^T) 

The correction d8 is likewise composed of two parts, namely, 

dd = dd + ^(Ai + dT+ JaT) 
Further, we have simply rfJ' = 3d' and 

df = dT+ ^aT'— ^tt' 

but, as we may neglect the error in the rate of the chronometer 
for the brief interval between the observation of the moon and 
the star, we can take 5a 7'' = 5a T, and, consequently, 

df=dT'+ d/iT—da' 

When the substitutions here indicated are made in (432), we 
obtain the expression 

x = a/^L + 15f.3a—^^^-^'dd — (15/— a)^r 

sin C 

— 15/'.^a'+?^^.o*J'+15/.^T' 

sin C 


sin C sin C 

_ [15 (/_/')- a] 3A r + «i"y-OBin^' g^ (433) 

sm C sm C 


in which the following abbreviations are used : 

^ cos^ cosq cos^'cosj^ 

sin C Bin C 

^ sin flr 

and in the coefficient of 8(p we have put A = A'. 

By the aid of this equation we can now trace the effect of 
each source of error. 

1st The coefficients of 55, 55', 5;r, dip have different signs for 
observations on different sides of the meridian, and therefore 
the errors of declination, parallax, and latitude will be elimi- 
nated by taking the mean of a pair of observations equidistant 
from the meridian. 

2d. The star's declination being nearly equal to that of the 
moon, we shall have very nearly /=/', and the coefficient of 
5A7'\\dll be = a; and since to find £^L we have yet to divide 
the equation by a, it follows that an error in the assumed clock 
correction produces an equal error (but with a different sign) in 
the longitude, as in the case of meridian observations. 

3d. An error 57^ in the observed time of the moon's transit 
produces in the longitude the error 



a I 


The mean of the values of a for tvvo observations equidistant 
from the meridian is Xf. The mean effect of the error 8T \a 



which is the same as in the ease of a meridian observation. 

The effect of an error 8T' in the observed time of the star's 
transit is 


and for two observations equidistant from the meridian, the star 
being in the same parallel as the moon, the mean effect is 

'A or 


also the same as for a meridian observation. 


4th. An error 3S in the tabular semidiameter is always elimi- 
nated in the case of meridian observations when they are com- 
pared with observations at another meridian, since the same 
semidiameter is employed in reducing the observations at both 
meridians. But in the case of an extra-meridian observation the 
effect upon the longitude b 

dS dS 

a sinC ^ cos d cosq — p sinq 

and in the mean of two observations equidistant from the 
meridian, the values of q being small, it is 

(1+2 sin* } q) nearly. 

X cos d cos q X cos d 

For a meridian observation the error will be 


The error in the case of extra-meridian observations, therefore, 
remains somewhat greater than in the case of meridian ones, the 
excess being nearly 


>lC08 d 

which, however, is practically insignificant ; for we have not to 
fear that dS can be as great as 1", and therefore, taking q = 15°, 
8 = 30°, and >l = 0.4, which are extreme values, the difference 
cannot amount to O'.l in the longitude. 

5th. The ciTor 8a of the tabular right ascension of the moon 
produces in the longitude the error 


and from the mean of two observations equidistant from the 
meridian, the error is 

15 da 

as in the case of the meridian observation. 

The error 8a' in the star's right ascension produces the error 

when the star is in the same parallel as the moon. 


From this discussion it follows that, by arranging the observa- 
tions symnieirkaUy with respect to the meridian, the mean resuh 
vn\\ be liable to no sensible errors which do not equally affect 
meridian observations. But for the large culmination error in 
the case of the moon (Art. 236), which equally affects extra- 
meridian observ^ations, the latter would have a great advantage 
by diminishing the effect of accidental errors. But the probable 
error of the mean of two obsen^ations equidistant from the 
meridian, seven threads being employed, will be, by (422), 

..= ^[(0.091).+ (2:1^+ ^] = 0..10 

and that of a single meridian observation, even where only one star 
is compared with the moon, is, by the same formula, = O*.!!. When 
we take into account the extreme simplicity of the computation, 
the method of moon culminations must evidently be preferred; 
and that of extra-meridian observations will be resorted to only 
in the case already referred to (Art. 239), where the traveller 
may wish to determine his position in the shortest possible time 
and A\nthout waiting to adjust his instrument accurately in the 

Example. — ^At the TJ. S. Naval Academy, 1857 May 9, I ob- 
serv^ed the following transits of the moon's second limb and of 
a Scorpiij at an approximate azimuth of 10° East, ^vith an Ertel 
universal instrument of 15 inches focal length : 

Chronometer. Lerel. CoUim. 

D II Limb. T = 16* 11- 30-.17 b = + 2".2 c = 0.0 ) Vertical circle 
tr Scarpa T = 16 27 49 .83 b' = + 2 .2 ) left. 

These times are the means of three threads. The chronometer 
correction, found by tmnsits of stars in the meridian, was 
— 55"* 9-.16 at 13* sidereal time, and its hourly rate — 0'.32. The 
assumed latitude and longitude were 

SP = 38^ 58' 53".5 X' = 5* 5- 55* 

The star's place was 

a' = 16* 12- 31'.90 ^' = — 25<^ 14' 58".5 



We first find the sidereal times of the observations of the 
moon and star respectively, and the Greenwich mean time of 
the observation of the moon : we have 

aT = — 65" 9-.89 
r + A T = 15* 16~ 20*.28 
L'= 5 5 55. 

Gr. sidereal time = 20 22 15 .28 
Sid. time Gr. moon = 3 8 58.91 

Sidereal interval = 17 13 16.37 
Red. to mean time = — 2 49 .28 

Or. mean time = May 9, 17* 10* 27*.09 

Hence from the Ephemeris wo find 

a = 15* 54~ 45'.32 
Aa = 2-.1135 
S = 14' 47".2 

Ar'=— 55~ 9*.97 
T' + aT'=15»32-39'.86 


24*' 42' 54".4 
54' 9".2 

By (426) we find 

A = — 

log sin q 


r sin C 

9^ 40' 51".0 
64^ 19'.5 
+ 48.8 
— 2.1 
65 6.2 

log sin g* 


9*' 57' 14".8 
64^ 54'.1 

— 2.1 
64 52.0 

For the latitude f we find, from Table in., 

log p = 9.9994 
and then, by (427), we find 

sp— sp'=ll'15" 



= _ 90 40' 5r.o 

— -r^ = — 16 24 .4 

/Mr(f — 

sin C 
f ') sin 1" sin A' 

sin C 


2 .0 


1 .0 

tan Ci 

^= — 9 57 18 .4 

il' = — 9^ 57' 14".8 

sin C/ 


1 .0 

tan Ci' _____ 

^i' = — 9 57 15 .8 



.r = — 2".6 

By (429), (430), and (431), we find 

log ;i = 9.72175 log /9 = n9.10266 a = 0.6064 

Ai= — :::^ = --5M4 

If we wish to see the eftect of all the sources of error in this 
example, we find, by (433), 

0.5054 aL = — 2".6 - 14.96 Aa — 0.16 AA -f- 14.45 AT— 14.82 Sr — 0.36 A^T 

+ 14.82 Aa' + 0.16 AS' + 1.11 AS — 0.001 Air -f 0.002 d# 

The proper combination of obsci'vations is supposed to eliminate, 
or at least reduce to a minimum, all the errors except that of the 
moon's right ascension as given in the Ephemeris. In practice, 
therefore, it will be necessary to retain the term involving ia. 
Thus, in the present case we take only 

0.5054 Ai = — 2".6 — 14.96 da 

A second observation on the same day at an azimuth 10** 

west gave 

0.5458 Ai = — 5".7 — 14.92 da 

The elimination of the errors of declination requires that we 

take the arithmetical mean of these equations; whence we have, 


Ai = — 7'.89 — 28.43 ^a 


242. The hour angle (i) of the moon may be computed from 
an observed altitude, the latitude and declination being known, 
and hence with the local sidereal time of the obser\'ation (=0; 
the moon's right ascension by the equation a = — t^ with 
which the Greenwich time can be found, as in Art 234, and, 
consequently, also the longitude. 

The hour angle is most accurately found from an altitude 
when the observed body is on the prime vertical, and more 
accurately in low latitudes than in high ones (Art. 149). Tliis 
method, therefore, is especially suited to low latitudes. 

The method maybe considered under two forms: — (A) that in 
which the moon's absolute altitude is directly observed and 


employed in the computation of the hour angle ; and (B) tliat in 
which the moon's altitude is compared differentially with that of 
a neighboring star, — i.e. when the moon and a star are observed 
either at the same altitude, or at altitudes which differ only by a 
quantity which can be measured with a micrometer. 

243. (A.) Bif the moon's absolute altitude. — This method being 
practised only with portable instruments, it would be quite 
superfluous to employ the rigorous processes of correcting for 
the parallax, which require the azimuth of the moon to be given. 
The process of Art 97 will, therefore, bo employed in this case 
with advantage, by which the observed zenith distance is reduced 
not to the centre of the earth, but to the point of the earth's 
axis which lies in the vertical line of the observer, and which 
we briefly designate as the point 0. Let 

C" = the observed zenith distance, or complement of the 

observed altitude, of the moon's limb| 
© = the local sidereal time, 
L'= the assumed longitude, 
Ai/ = the required correction of L', 
L = the true longitude =z L* -\- ^L. 

Find the Greenwich sidereal time © +-L', and convert it into 
mean time, for which take from the Ephemeris the quantities 

9 = the moon's declination, 

r = " eq. hor. parallax, 

S = " semidiameter. 

Let S' be the apparent semidiameter obtained by adding to S 
the augmentation computed by (251) or taken from Table XEE. 
Let r be the refraction for the apparent zenith distance f " ; and 


C' = C" + r d= S' (434) 

Let TTi be the corrected parallax for the point 0, found by (127), 
or by adding to tt the correction of Table XDI. (which in the 
present application will never be in error 0".l) ; and put 

a, = a + e» ffj sin ^ cos J | 

C, = C'-^,sinC' } (^^^) 

in which log e*= 7.8244. 


The hour angle (which is the same for the point as for the 
centre of the earth) is then found by (267), Le. 

Binie^Jf"°^[^' + (^-''')^""^t-'-(^-^-)J\ (436) 

\ \ cos f COS ^j / 

after which the moon's right ascension is found by the formula 

a = e — < (437) 

and hence the Greenwich time and the longitude as above stated. 
But since we have taken 3 for an approximate Greenwich time 
depending on the assumed longitude, the first computation of i 
will not be quite correct ; a second one with a corrected value 
of 3 will give a nearer approximation ; and thus by successive 
approximations the true value of t and of the longitude will at 
last be found. 

But instead of these successive approximations we may obtain 
at once the coiTCction of the assumed longitude, as follows. We 
have taken 3 for the Greenwich time © + i', when we should 
have taken it for the time © + i' + Ai. Hence, putting 

/9 = the increase of ^ in a unit of time, 

it follows that o requires the correction ^^L; and therefore, by 
(51), the correction of the computed hour angle will be 

cos d tanq 

in which q is the parallactic angle. Since a = — /, the com- 
puted right ascension requires the correction (in seconds of time) 

15 cos J tanq 
Therefore, if we put 

X =z the increase of a in a unit of time, 

the computed rfreenwioh time and, consequently, also the longi* 
tude derived from it requires the correction 

15 >l cos ^ tang 



Hence, denoting the longitude computed from the right ascen- 
Bion a = — thj i", we have 

True longitude = i' + Ai = Z" — 




Ai = 


1 + 


15 il 

sec d cot jT 

If we denote the denominator of this expression by 1 + a, we 
shall have, by (18), 


15>l\ sin^ tant / ^ ^ 

and then 

^''"■^' i = i'4.Ai (439) 

Ai = 

1 + a 

' Example. — ^At the U. S. Naval Academy, in latitude (p = 88® 
58' 58" and assumed longitude i ' = 5* 6"* ()•, I observed the 
double altitude of the moon's upper limb with a sextant and 
artificial horizon as below : 




AvnoMd X' 
Approz. Or. time 

1S49 May 2. — Moon east of the meridian. 
10* 14» 21*.6 
4 41 0.0 

6 88 21.6 
6 6 0. 

10 88 21.6 

(For which time we take v, S, and 
8 from tlie Nautical Almanac.) 

a =+ 8«»47'47''.6 
«>Viiio ^ oca a » -H 14 .1 

a« *- + 8 48 1 .7 

Mean of 6 oba. 2 p » 64o 40^ 0^ 
Indexoorr. ofieztanta — 14 57 

App. alt. D 

Alt. Therm. 


63° F. 
660 F. 
5— 16'16".4 
Air (Tab. xn.)s 

+ 8 .1/ 

Ajr (Tab. XnL)» + 4 .4 Ivi 
n » 66 7 .5) 


2 )64 26 8 

. 82 12 81 .6 

' 67 47 28 .6 

> + 1 80 i) 

' + 16 24 .6 

> 58 4 23 .9 

47 88 .1 

> 57 16 45 .8 

With these values of d^, Ci» »»<! 9 = 38° 58' 53", we find, by (436), 

/ = — 3* 19- 53'.64 

The sidereal time at Greenwich mean noon, 1849 May 2, was 
2*41"'7*.98; whence 

e= 8»16-14-.61 
a = 11 36 8.25 

Vol. L— 24 


Corresponding to tliis right ascension we find by the honrij 
Ephemeris the Greenwich mean time, and hence the longitude 
i", as follows: 

Greenwich mean time = 10* 89* 48*.7 
Local '* " = 5 33 21.6 

X"= 5 6 27.1 
L''—L' = + 27M 

By the hourly Ephemeris we also have for the Greenwich time 

10* 39- 48'.7, 

Increase of a in 1* = ; = -f 2'.014 

« ^ in 1- = ^9 = + 10".01 

and hence, by (438) and (439), 

a = — 0.3317 Ai = + 40-.6 

i = i' + Ai = 5* 6* 40*.6 

244. The result thus obtained involves the errors of the 
tabular right ascension and declination and the instrumental 
error. The tabular errors are removed by means of observations 
of the same data made at some of the principal observatories, as 
in the case of moon culminations. The instrumental error will 
be nearly eliminated by determining the local time from a star 
at the same altitude and as nearly as possible the same declina- 
tion ; for the instrumental error will then proiiuce the same 
error in both and /, and, therefore, will be eliminated from 
their difForcnce — t = a. The error in the longitude will 
then be no greater than the error in 0. But to give complete 
effect to this mode of eliminating the error, an instrument, such 
as the zenith telescope, should be employed, which is capable of 
indicating the same altitude with great certainty and does not 
involve the errors of graduation of divided circles. A very 
ditforcnt method of observation and computation must then be 
resorted to, which I proceed to consider. 

245. (B.) Bi/ equal aliltmlcs of the moon and a star^ observed with 
(he zauth telescope. — The reticule of this instrument should for 
these observations be provided with a system of fixed Iiorizontal 
threads: neveiiheloss, we may dispense with them, and employ 
only the single movable micrometer thread, by setting it suc- 
cessively at convenient inten'als. 


Having selected a well determined star as nearly as possible 
in the moon's path and differing but little in right ascension, a 
preliminary computation of the approximate time when each 
body will arrive at some assumed altitude (not less than 10°) 
must be made, as well as of their approximate azimuths, in 
order to point the instrument properly. The instrument being 
pointed for the first object, the level is clamped so that the 
bubble plays near the middle of the tube, and is then not to be 
moved between the observation of the moon and the star. After 
the object enters the field, and before it reaches the first thread, 
it may be necessary to move the instrument in azimuth in order 
that the transits over the horizontal threads may all be observed 
without moving the instrument during these transits. The times 
by chronometer of the several transits are then noted, and the 
level is read off. The instrument is then set upon the azimuth 
of the second object, the observation of which is made in the 
same manner, and then the level is again read off. This com- 
pletes one observation. The instrument may then be set for 
another assumed altitude, and a second observation may be taken 
in the same manner.* Each observation is then to be separately 
reduced as follows : Let 

1, t', H'y &c. = the distances in arc of the several threads 

from their mean, 
m, m' = the mean of the values of i for the observed 
threads, in the case of the moon and star 
/, r= the level readings, in arc, for the moon and 
0, 0'= the mean of the sidereal times of the observed 
transits of the moon and star; 

then the excess of the observed zenith distance of the moon's 
limb at the time above that of the star at the time ©' isf 

the quantities m and I being supposed to increase with increasing 
zenith distance. 

* The same method of obseryation may be followed with the ordinary universal 
instrument, but, as the level is generally much smaller than that of the zenith tele- 
scope, the same degree of accuracy will not be possible. 

f Wheu the micrometer is set successively upon assumed readings, m and m' will 
be the means of these readings, converted into arc, with the known value of the screw. 


A](flo, let 

o, ^, t, C, A, J = the E. A., decl., hour angle, geocentric 

zenith distance, azimuth, and parallactic 
angle of the moon's centre at the time 


o', d', Ify C', A' J g[ = the same for the star at the time O'; 

7tj S = the moon's equatorial hor. parallax and 
X = the increase of a in 1* of sid. time ; 

fi :=. " ^ « « <« 

^ = the latitude ; 
X' = the assumed longitude; 
A-L = the required correction of Jj'; 

The quantities a, d, ;r, and S are to be taken from the Ephemeris 
for the Greenwich sidereal time + i ' (converted into mean 
time) ; a and 8 being interpolated with second differences by the 
hourly Ephemeris. Then the required correction of the longi- 
tude will be found by comparing the computed value of ^ with 
the observed value. For this purpose we first compute f and {^' 
\vith the greatest precision, and also A and q approximately. H 
the differential formula of the next article is also to be computed, 
A' and j' will also be required. The most convenient formulae 
will be — 

For the moon. For the star. 

t^e — a f=e'_a' 

tan M=t&n d sec t ^ -^i. - ( tanalf'= tan S' sec f 

8m^co8(c> — M) > , . I < ,, sin^'cosCcp — M') 

cos C = — ( decimals; ) cosC == - 

8in M J K sin M' 


cos -4= tan (^ — Jf)cotZ \ I cos A' = tan (^ — Jf')cotC' 

taniV=eot^ cos^ / y^iW^ four ) tan i\r'=: cot ^ cost' 

tan f sin JV i HopiTnAln- i . # tan f sin ^V 

tan <7 = \ decimals , j tan ^ = 

cos {pj^N)) \ cos (^' + X") 

The zenith distance f thus computed will not strictly correspond 
to the time unless tlic assumed longitude is correct. Let its 
true value be ^ + d^. Also put 

Cj =r the observed zenith distance of the moon's limb, 
Cj' = the observed zenith distance of the star, 
r^r' = the refraction for C, and C', 



Cj =c/+m — m' + Z — r 
Putting then 

C"=Ci + r = C +m — m'+/ — r+(r — O 
and, by Art (136), 

: (441) 

r = (^ — 9') 00s A sinjp=/t)8in7r8in(C" — r) 

k=zpz^ /gq: iijpT^ /S) sin |) sin S 

the 1 1 \ sign being used for the moon's ^ i \ limb, we 


This equation determines rf^. We have, therefore, only to 
determine the relation between rff and Ai. Now, we have taken 
a and d for the Greenwich sidereal time @ -\- L'^ when we should 
have taken them for the time © + i ' + aZ ' : hence 

a requires the correction k^L 
a " « i9Ai 

t " « — AAi> 

and then, by (51), 

dZ := — cos q . p^L — sin jr cos ^ . 15 >l ^L 

Hence, putting x = — rf^, or 

x = Z — C' + k 
and a = \blBmq cos ^ + i^ ^^ ? 

we have i^L = — L = L' + ^L 


The solution of the problem, upon the supposition that all the 
data are correct, is completely expressed by the equations (440), 
(441), and (442). 

246. The quantity x is in fact produced not only by the error 
in the assumed longitude, but also by the errors of observation 
and of the Ephemeris. In order to obtain a general expression 


in which the eflfect of every source of error may be represented, 

T, T'= the chronometer times of observation of the 
moon and star, 
A r = the assumed chronometer correction, 
dTjdT= the corrections of T and T for errors of 
d^T = the correction of a T, 
da, dd, dn, dS = the corrections of the elements taken from 

the EphemeriS; 
^^ == the correction of the assumed latitude. 

K, when the corrected values of all the elements are substituted, 
^, f ', k become ^ + rf^, ^' + ^C'* ^ + ^^'> instead of the equation 
f " —{^ + dC) = k we shall have 

C"+ (?:' — (C + d:) = k + dk 
and hence 

x = — d: + d:'^dk (448) 

and we have now to find expressions for rf^, df ', and dk in terms 
of the above corrections of the elements. 
Taking all the quantities as variables, we have 

dZ = 15 sin q cos d dt — cos q dd -\- cos A df 
<fC'= 15 sin 5^ cos ^'rf^ — cos^c/^'-f qo^A* dtp 

Since ^ = J* + a J* — a, we have 

dt = dT -\- di^T^da 

where dT^and rf^T^may be exchanged for 5 7" and 5a J*, but da is 
composed of two parts: Ist, of the actual correction of the 
Ephemcris; and 2d, of ^(aZ + dT -{■ d^T) resulting from our 
having taken a for the uncorrected time : hence we have 

The correction d8 is also composed of two parts, so that 

dd = dd + fi(j:iL + dT+ ^aT) 

Further, we have simply d8^ = Sd\ and 

df=dT+ dAT^da' 

in which 5a T at the time T' is assumed to be the same as at the 


time T^ an error in the rate of chronometer being insensible in 
the brief interval between the observations of the moon and the 

Again, we have, from (441), 

cos p dp = p cos n sin (C" — z') <i:c + /> sin it cos (C" — r) <^» " 

dk = dp z^: dS 

or, with sufficient accuracy, 

dk = sin C' dn :^ dS -{- %m 'k cos C' dZ' 

Now, substituting in rff and rfi^' the values of rf/, rf5, &c., and then 
substituting the values of rff and rff ' thus found, in (443), together 
with the value of rfA, we obtain the final equation desired, which 
may be written as follows :* 

x = a£kL-^f .Ha + co&q.dd —(f^ a)dT 

— mf.da' — m cos q* dd' + mf . dT 

— (cos A — m cos A') dip 

where the following abbreviations are employed : 

/ = 15 sin q cos d f^ =z\h sin g[ cos d' 

a = Xf -\- fi cosq wi = 1 — sin tt cos C' 

Having computed the equation in this form, everj' term is to 
be divided by a, and then aZ/ will be obtained in terms of z and 
all the corrections of the elements. 

A discussion of this equation, quite similar to that of (433), 
will readily show that the observations will give the best result 
when taken near the prime vertical and in low latitudes, and, 
farther, that the combination of observations equidistant from 
the meridian, east and west, eliminates almost wholly errors of 
declination and parallax and of the chronometer correction. 

Example.! — At Batavia, on the 11th of October, 1853, Mr. 
Db Lange, among other observations of the same kind, noted 
the following times by a sidereal chronometer, when the moon's 

* The formula (444) is essentially the same as that giyen by Oudemans, Aatronom. 
Journal, Vol. IV. p. 164. The method itself is the BUggestion of Professor Kaiser 
of the Netherlands. 

f Astronomical Journal, Vol. IV. p. 165. 

892 LONGirUDE. 

lower limb and 86 Octpricomi passed the same fixed horizontal 

i^iireflifls * 

T = 0* 38- 8'.62 T' = 0» 49- 68'.77 

The difierence of the zenith distances mdicated by the level 


Z — r = + 2".0 

The chronometer correction was aT= + 1"* 8*. 82, and the rate 
in the interval T' — J* was insensible. 

The assumed latitude was ^ = — 6® y 57".0 

" longitude " i' = -- 7* 7- 37'.0 

We have 

8 = 0* 39- 11-.94 8'= 0* 50- 67'.09 

For the Greenwich sid. time © + i' = 17* 81- 34*.94,or mean 
time 4* 10- 67*.00, we find, from the Nautical Almanac, 

a — 21» 12- b'Ab 

X — + 0'.0387 

d — 20° 55' 8".9 

i9 — + 0".1440 

TT — 57' 51".4 

a'= 21* 20- 22'.45 

S — 15' 47".8 

a'— — 22° 26' 80".5 

)utation by (440) gives 

C — 52° 11' 49".44 

C— 53°13'57".30 

A — 68° 14'.4 

^'— 66°30'.6 

q — 81° 18'.9 


ble TTT. we find 

ip q>'— 2' 27" 

log p — 9.999983 

Since the same fixed threads were used for both moon and star, 
we have m = m', and hence also sensibly r = r^; therefore, by 
(441), we find 

C" = 53° 13' 59".30 r = — 54".5 p = 46' 21".25 

C — C"=— 62' 9".86 A=: + 62'9".17 

Hence, by (442), 

x = ^ 0".69 a = + 0.5575 aZ = — 1'.24 

The longitude by this observation, if the Ephemeris is correct, 

is therefore 

i =^ Z' + aZ :^ — 7* 7- 38-.24 


If we compute all the terms of (444), we shall find 

Ai = — K24 — 24.84 ^a — 0.27 M + 28.84 ^T— 24.24^2" — 0.44 ^aT 

+ 24.28 aa'+ 0.29 ^a'+ 1,79 dS+ IMdn — 0.04^^ 

This shows clearly the effect of each source of error; but in prac- 
tice it will usually be sufficient to compute only the coefficients 
of da and dd. In the present example, therefore, we should take 

Ai = — 1-.24 — 24.84 da — 0.27 M 

which will finally be fully determined when da and 88 have been 
found from nearly corresponding observations at Greenwich or 


247. The distance of the moon from a star may be employed 
in the same manner as the right ascension was employed in 
Arts. 229, &c., to determine the Greenwich time, and hence the 
longitude. K the star lies directly in the moon's path, the 
change of distance will be even more rapid than the change of 
right ascension ; and therefore if the distance could be measured 
with the same degree of accuracy as the right ascension, it would 
give a more accurate determination of the Greenwich time. 
The distance, however, is observed with a sextant, or other re- 
flecting instrument (see Vol. 11.), which being usually held in 
the hand is necessarily of small dimensions and relatively infe- 
rior accuracy. Nevertheless, this method is of the greatest im- 
portance to the travelling astronomer, and especially to the 
navigator, as the observation is not only extremely simple and 
requires no preparation, but may be practised at almost any 
time when the moon is visible. 

The Ephemerides, therefore, give the true distance of the 
centre of the moon from the sun, from the brightest planets, and 
from nine bright fixed stars, selected in the path of the moon, 
for every third hour of mean Greenwich time. The planets em- 
ployed are Saturn, Jupiter, Mars, and Venus. The nine stars, 
known as lunar-distance stars, are a Arietis, a Tauri {Aldebaran\ 
P Geminorum (Pollux), a Leonis (Regulus), a Virginis (Spied), 
a Scorpii (Ajitares), a Aquite (Aliair), a Piscis Australis (Fomal- 
hau()j and a Pegasi (Markab). 

The distance observed is that of the moon's bright limb from a 

Fig. 29. 

star, from the estimated centre of a planet, or from the limb of 
the BUn. The apparent distance of the moon's centre from a st«r 
or planet is found by adding or subtracting the moon's apparent 
(augmented) semidiameter, according as the bright limb is nearer 
to or farther from the star or planet than the centre. The ob- 
served distance of the aun and moon is always that of the nearest 
limbs, and therefore the apparent distance of the centres is fotuul 
by adding both seniidiameters.* 

The apparent distance thus found differs from the (nte (geo- 
centric) distance, in eonaequeuce of the parallax and refraotioo 
which affect the altitudes of the objects, and consequently also 
the distauee. The true distance is therefore to be obtained by 
computation, the general principle of which may be exhibited tn 
a simple manner as follows. Let Z, Fig. 29, 
be the zenith of tlie observer, M' and .S" the ob- 
served places of the moon and star, S[M' the 
parallax and refraction of the moon, .*iS' tho 
refraction of the star, so that M and S are tlie 
geocentric places. The apparent altitades of 
the objects nia^v either be measured at the same 
time as the distance, or, the local time bving 
known, theymaybe computed (Art. 14). The apparent zenith dis- 
tances, and, consequently, also the true zenith distances, an? there- 
fore kno«Ti. Ill the triangle ZM'S' there are known the threo 
sides, ^/'.Sf' the apparent distance of the objects. 2J/' the apfiarunt 
zenith distance of the moon, and Z8' the apparent zenith distaoca 
of the star: from which the angle 2 is computed. Then, in tho 
triangle ZMS there are known the sides, ZM the moonV true 
zenith distjnice, and ZS the star's true zenith distance, and the 
angle Z; from which the required true distance J/.V is coinputcd. 
In this elementary explanation the parallax and refrnctiou of 
the moon are supposed to act in the same vertical circle ZM, 
whereas parallax acts in a circle drawn through the moon and 
the geocentric zenith (Art, 81 ). while refraction acta in tho vertical 
circle drawn through the astronomical zenith. Again, when the 
moon, or the sun, is observed at an altitude less tlmn 50°, it ia 
necessary to take into account the distortion of tho disc prodaced 

* We mtij also observe tho tlUlnnee fram the liroh of k plkoat. prorided Ih* tt%- 
tant lelcsoape is of suHicieai power la gtre ibe pUiiel B vcll-defiDeil dUr ; and iIm 
pUoet'a scmidiaincter in then also lo b« added or aubtncted. 


by refraction if we wish to compute the true distance to the 
nearest second of arc (Art. 133). These features, which add 
very materially to the labor of computation, cannot be over- 
looked in any complete discussion of the problem. 

Simple as the problem appears when stated generally, the 
strict computation of it is by no means brief; and its importance 
and the frequency of its application at sea, where long computa- 
tions are not in favor, have led to numerous attempts to abridge 
it. In most instances the abbreviations have been made at the 
expense of precision ; but in the methods given below the error 
in the computation will always be much less than the probable 
error of the best observation with reflecting instruments : so that 
these methods are entitled to be considered as practically perfect. 

With the single exception of that proposed by Bessel,* all the 
solutions depend upon the two triangles of Fig. 29, and may be 
divided into two classes, rigorous and approximative. In the 
rigorous methods the true distance is directly deduced by the 
rigorous formulae of Spherical Trigonometry ; but in the approxi- 
mative methods the difference between the apparent and the 
true distance is deduced either by successive approximations or 
from a development in series of which the smaller terms are 
neglected. Practically, the latter may be quite as correct as the 
former, and, indeed, with the same amount of labor, more 
correct, since they require the use of less extended tables of 
logarithms. I propose to give two methods, one from each of 
these classes. 

A. — The Rigorous Method, 

248. For brevitj', I shall call the body from which the moon's 
distance is observed tke sun^ for our formulae will be the same 
for a planet, and for a fixed star they will require no other 
change than making the parallax and semidiameter of the star 

* Attron. Nach, Vol. X. No. 218, and Attron. UnUrsuehungfny Vol. II. Bessel's 
method requires a different form of lunar Ephemeris from that adopted in our 
Nautical Almanacs. But eren with the Ephemeris arranged as he proposes, the 
computation is not so brief as the approximative method here giyen, and its supe- 
riority in respect of precision is so slight as to giye it no important practical 
advantage. It is, howeyer, the only theoretically exact solution that has been given, 
and might still come into use if the meaturement of the distance could be rendered 
much more precise than is now possible with instruments of reflection. 


Let US suppose that at the given local mean time Tihe obser- 
vation (or, in the ease of the altitudes, computation) has given 

d"= the apparent distance of the limbs of the moon and 

h' = the apparent altitude of the moon's centre, 
H'=z the apparent altitude of the sun's centre, 

and that in order to compute the refraction accurately the 
barometer and thermometer have also been observed. For the 
Greenwich time corresponding to T, which will be found with 
sufficient accuracy for the purpose by employing the supposed 
longitude, take from the Ephemeris 

8 = the moon's semidiameter, 

S = the sun's « 

then, putting 

d'= the apparent distance of the centres, 
^ == the moon's augmented semidiameter, 
= a + correction of Table XII. 
we have 

upper signs for nearest (inner) limbs, lower signs for farthest 
(outer) limbs. 

But if the altitude of either body is less than 50°, we must 
take into account the elliptical figure of the disc produced by 
refraction. For this purpose we must employ, instead of s' and 
Sy those seraidiameters which lie in the direction of the lunar 
distance. Putting 

q = ZM'S\ Q = ZS'M' (Fig. 29) 

AS, ^S = the contraction of the vertical semidiameters of the 
moon and sun for the altitudes h' and H\ 

the required inclined semidiameters will be (Art 133) 
s' — AS cos' q and S — aS cos* Q 

The angles q and Q will be found from the three sides of the 
triangle ZM'S\ taking for d' its approximate value d" ±: s' ±i S 
(which is sufficiently exact for this purpose, as great precision in 
q and Q is not required), and for the other sides 90° — h' and 
90° - H\ K we put 

m = i(A'+ H'+d') 


we shall have 

. ,- coBm8in(m — IE') . ,,^ cosmsinCm — A') ,..^^ 

8mMff = r—^ r; — - Bm«}C= r—jj^ -£^ (445) 

sin d' cos A' sin d' cos H 

and then the apparent distance by the formula 

d' = d" ± (s' — A5 co8« q)±(^8— aS COS* Q) (446) 

We are now to reduce the distance to the centre of the earth. 
We shall first reduce it to that point of the earth's axis which 
lies in the vertical line of the observer. Designating this point 
as the point 0, Art. 97, let 

d^, hjj H^ = the distance and altitudes reduced to the point 


r, R = the refraction for the altitudes h' and H', 
Xj P z= the equatorial hor. parallax of the moon and 

The moon's parallax for the point will be found rigorously 
by (127), but with even more than sufficient precision for the 
present problem by adding to k the correction given by Table 
XIII- Denoting this correction by A;r, we have 

Ai = A'— r + TTi cos (A' — r) J?i = ^' — 72 + Pcos (J' — i?) (447) 

The parallax P is in all cases so small that its reduction to the 
point is insignificant. 

If, then, in Fig. 29, M and S represent the moon's and sun's 
places reduced to the point 0, and we put 

Z = the angle at the zenith, MZ8, 

we shall have given in the triangle M'ZS' the three sides 
rf', 90° — A', 90° — i?', whence 

, , - cos } (A' + J' + d') cos } (A' + ^' — d') 

cos' } z = ^^ ■ ■ > ■ 

cos A' cos H' 

and, then, in the triangle MZS we shall have given the angle Z 
with the sides 90° — Aj and 90° — -Hj, whence the side MS = d^ 
will be found by the formula [Sph. Trig. (17)], 

sin* } di = cos' } (Aj + Jli) — cos Ai cos J?i cos' } Z 


To simplify the computation, put 

m = i(h' + H' + d!) 
then the last formula, after substituting the value of Z, becomes, 

sm' J <?i = cos' J (Aj -f J?i) cos m cos (m — <f ) 

COB A' cos W 

Let the auxiliary angle M be determined by the equation 

sin* if = ^Q« ^1 <^Qg -^1 . cos m cos (m — dT) 

cos A' cos ^' cos' i (Ai + IQ ^ 

then we have* 

sin } di = cos J (^ + ^i) cos I£ (449) 

Finally, to reduce the distance from the point to the centre 
Fig. 30. of the earth, let P (Fig. 30) be the north pole of 

the heavens, M^ the moon's place as seen from the 
point 0, M the moon's geocentric place, S the 
^^ sun's place (which is sensibly the same for either 
point). The point being in the axis of the 
celestial sphere, the points JIfj and -3f evidently lie 
in the same declination circle PM^M, Hence, 

d = the geocentric distance of the moon and sun = SM^ 

d = the moon's geocentric declination = 90° — PAf, 

^j =^ the declination reduced to the point = 90® — PJfj, 

J = the sun's declination = 90° — PS^ 

we have, in the triangles PMS and M^MS^ 

^ ^ cos J. — cos (d, — <5) cos d sin J — sin d cos d 

cos PMS = ^-^^ = : 

sin ((Jj — d) sin d cos d sin d 

We may put cos {d^ — 5) — 1, and, therefore, 

sin (^, — ^) , . . . , jx 
cos d^ — cos d = ^^-^ (sm J — am ^ cos a) 

cos d 

* This transformation of the formulas is due to Borda, Description et tuagt du ctrtU 
de rffltxion. 


and since d — d^ is very small, we may put cos d^ — cos d = 
sin {d — rfj) sin rfj, and hence, very nearly, 


(sin J sin d \ 

sin c?j tan d^ I 

cos d 

Substituting the value of ^^ — 3 from (122), 

^ J A ' / sin J sin ^ \ ,._^ 

d — d^ = Ai: Bin ^l ) (450) 

\ sin d^ tan d^ I 

in which y is the latitude of the observer, and log A may be 
taken from the small table given on p. 116. The correction 
given by this equation being added to d^, we have the geocentric 
distance d according to the observation. 

To find the longitude, we have now only to find the Green- 
wich mean time Tq corresponding to rf, by Art. 66, and then 

L=T,^T (451) 

Example. — ^In latitude 35° N. and assumed longitude 150° W., 
1856 March 9, at the local mean time jT = 6* 14« 6*, the ob- 
served altitudes of the lower limbs and the observed distance 
of the nearest limbs of the moon and sun were as follows, cor- 
rected for error of the sextant : 

A" = 52° 34' 0" -ff" = 8° 56' 23" d" = 44° 36' 58".6 

The height of the barometer was 29.5 inches, Attached therm. 
60° P., External therm. 58° P. 

I shall put down nearly all the figures of the computation, in 
order to compare it with that of the approximative method to be 
given in the next article. 

Ist. The approximate Greenwich mean time is 5* 14~ 6' + 10* 
= 15* 14* 6*, with which we take from the American Ephemeris 

s = 16' 23".l IT = 60' 1".9 ^ = + 14° 19' 

8=l& 8".0 P= 8".6 J = — A? 3' 

2d. To find the apparent semidiameters, we first take the 
augmentation of the moon's semidiameter from Table XTT., 
= 14".0, and hence find 



Then to comput<3 the contraction produced by refraction we find 
from the refraction table, for the given observed altitudes, the 
contractions of the vertical semidiameters (Art 132), 

A8 = 0".4 aS = r .6 

With the approximate altitudes and distance of the centres we 
then proceed by (445), as follows : 

d' = 45° l(y log ooseo d' 0.1498 log cosec d' 0.149S 

h' =52 51 log sec A' 0.2190 

ff' = 9 12 log see JJ' 0.0056 

HI =53 37 log COB m 9.7782 log cos m 9.7782 

m — ir'=44 26 log sin (m — IT') 9.8460 
m — A' = 46 log sin (» — *') aiOS 

9.9866 8.0646 

log sin }^ 9.9988 log sin } ^ 9.Q27S 

q= 169«66' Q=: 12? W 

log cos* (jr 9.9466 log cos* (^ 9.9800 

logA« 9.6021 logA5 0.9888 

9.6477 a962S 

Atco8«g= 0."4 A5cos»C= ^-2 

Hence we have, by (446), 

d"=440 86'58".6 
a' — AS cos* q =z 16 36 .7 
S—aScob'Q= 15 58 .8 

ef'=45 9 34.1 

3d. To find the apparent and true altitudes of the centres. — ^The 
apparent altitudes of tlie centres will be found by adding the 
contracted vertical semidiameters to the ob8er^'ed altitudes of the 
limbs. The apparent altitudes, however, need not be computed 
with extreme precision, provided tliat the difl[erenees between 
them and tlic true altitudes are correct; for it is mainly ui)on 
these difterenees that the difference between the apparent and 
true distance depends. 

The reduction of the moon's horizontal parallax to the point 
for the latitude 35° is, by Table XIIL, at: = 3".9; and hence 
we have 

rj = r + A?: = 60' 5".8 

and the computation of the altitudes by (447) is as follows : 


A" = 52*^34' 0" J" = 8° 56' 23" 

Vert, semid. = 16 37 Vert, semid. = 15 58 

h! = 52 50 37 J' = 9 12 21 

Table Il.r = 42 .7 E = 5 33 .6 

A' — r =52 49 54.3 ff'—B =9 6 47.4 

logjT, 3.55700 logP 0.9345 

log cos (A' — r) 9.78115 log cos (-ff'— R) 9.9945 

3.33815 0.9290 

r,cos(A'~r) =__36;j8^5 Pcos(^'— JK) = ^'.5 

Ai = 53^ 26' 12".8 H^= 9^ 6' 55".9 

4th. We now find the distance d^ by (448) and (449), as follows : 

d' = 45« 9'34".l 

A' = 52 50 37 log sec 0.2189683 

g'= 9 12 21 log sec 0.0056300 

m = 53 36 16 .1 log cos 9.7733154 

m — d' = 8 26 42 . log cos 9.9952654 

Ai = 53 26 12 .8 log cos 9.7750333 

H^= 9 6 55 .9 log cos 9.9944803 

2) 9.7626927 

i(hi+ -Hi) = 31 16 34 .4 log cos 9.9318007 9.9318007 

log sin M 9.9495457 log cos M 9.6583330 

idi=22 54 9.2 log sin J (?i 9.5901337 

di = 45 48 18 .4 

6th. To find the geocentric distance, we have, by (450), 
for f = 35°, 

log A 7.8249 d = ^Uo 19/ 

log IT 8.5565 J=— 4 3 

log sin f> 9.7586 

1.1400 1.1400 

log sin J n8.8490 log sin d 9.3932 

log cosec di 0.1445 log cot di 9.9878 

n0.1335 n0.5210 

— 1".4 — 3".3 

ci — ^= — 4".7 

d = 45*> 48' 13".7 

6th. To find the Greenwich mean time corresponding to rf. 

Vol. L— 2« 


and hence the lon^tude, according to Art. 66, we find an ap- 
proximate time {T) + i hj simple interpolation, and then the 
required time Tq = {T) + t + At, taking At from Table XX., 
with the arguments t and aQ {= increase of the logarithms in 
the Ephemeris in 3*), as follows : 
By the American Ephemeris of 1856 for March 9, we have 

r) = i5* 0- 0* 

(d ) = 45^ 40' 54" Q = 0.2510 

d =45 48 13 .7 

^ = 13 4 

7 19 .7 log — 2.6432 

At— 1 

log t = 2.8942 

To— 15 13 3 

T ~ 5 14 6 

£ = 9 58 57 

B. — The Approximative Method. 

249. I shall here give my own method (first published in the 
Astronomical Journal, Vol. II.), as it yet appears to me to be 
the shortest and most simple of the approximative methods 
toheii these are rendered siifficiently accurate by the introduction of all 
the necessary corrections. Its value must be decided by the im- 
portance attached to a precise result. There are briefer methods 
to be found in every work on Navigation, which ^vill (and should) 
be preferred in eases where only a rude approximation to the 
longitude is required. 

As before, let 

A', H' = the apparent altitudes of the centres of the moon 
and sun, 
ei" = the observed distance of the limbs, 
Sy S = their geocentric semidiameters, 
r, P = their equatorial horizontal parallaxes, 

5' = the moon's semidiameter, augmented by Table 

TTi =: the moon's parallax, augmented by Table XIII. 

We shall here also first reduce the distance to the point of 
Art. 97. The contractions of the semidiamcters prcKluced by 
refraction will be at first disregarded, and a correction on that 
account will be subsequently investigated. If then in Fig. 29, 
p. 394, jl/' and S' denote the apparent places, -Sf and S the places 
reduced to the point 0, we shall here have 


h! = W^ — ZM\ W = 90«> — ZS\ 

A,= 90*>— ZM, -Hi = 90°— ZSy 

and the two triangles give 

_ cos </, — sin Ai sin H^ cos d* — sin V sin W 
cos ^ = = ^^^— — — ^— ^— 

cos hx cos Hx cos A' cos ^' 

from which, if we put 

sin A, sin K. cos A, cos K. 
v(i = * -* n = ^ » 

sin A' sin -ff' cos A' cos IT 

we derive 

cos <i' — COS d^ = (l — n) cos d' + (n — m) sin A' sin H' (a) 

A<f = <fj— d' aA = Aj— A' AH=H'—ff^ (6) 

then we have 

cos (i' — cos<fj = 2 sin}A(f sin (4'-f }A<f) (c) 


cos(A^+aA) cos (H' — Ag) 
COS A' cos H' 

/ 2 sin t aA sin (A^+ j aA) \ / 28intAgsin(g^— iAg) \ 
\ "" cos A' P\ "^ cos J' ) 

2 sin t aA sin (A^ + t aA) 2 sin j Ag sin (IT^— t aJ) 
cos A' cos IT' 

, 4 sin t A A sin j Affsin (A^ + j a A) sin (H' — } aIT) 
"' cos A' cos -ET ^^ 


sin A' cos A, sin H' cos jH". — cos A' sin A, cos -EP sin H, 

n — m = ^ 

sin A' cos A' sin JET' cos H' 

Bubstitating in which the values 

2 sin A' cos A^ = sin (2 A' -f aA) — sin aA 
2 cos A' sin A, = sin (2 A' + aA) -f- sin a A 
2sin^'co8^j=sin(2ir'— Al?) + sinA-ff 
2 cos JT' sin H^ = sin (2 JT' — aJT) — sin aIT 

we find 

sinAg8in(2A^+AA) — sinAA8in(2-EP— aJT) 
^"^^ 2 sin A' cos A' sin jff' cos J' ^^^ 


Substituting (c), (rf), and (e) in (a), and at the same time, for 
brevity, putting 

. 2 sin i ^A sin (hf + J aA) 

^ cos A' 

^ _ sin aA sin (2 H' — Ag) 
* ~" 2 cos A' cos H' 

^ _ 2 sin i AJsin (g^— i A^ 
' ■" cos H' 

J. sin aH sin (2 A' + aA) 

*~~ 2 cos A' cos JET' 

we have 

2 8in}A<fsm(d'+jAcZ)=AiCOsd'+jBi + Cicos(f'+A— ^Cicos^i' CO 

This formula is rigorously exact ; but, since Arf is always less 
than 1°, it will not produce an error of O'M to substitute the arcs 
J Arf, J aA, &c. for their sines, or J Arf sin 1", i aA sin 1", &c. for 
sin J AC?, sin J aA, &c. ; and therefore we may write 

/^dHm(d'+i£id)===A^cosd'+Bi + CiCOBd'+I>i^AiCiBinV'co8d' (g) 
in which A^y B^^ Q, jDp now have the following signification : 

. sin (A' + i aA) 

cos A 

-, aA sin(2Jr'— aJT) 

cos h' 2 cos W 

A 7T 

Ci = - ----- . sin (J?'- J AlT) 

cos if' 

Aff sin (2 A' + aA) 

cos H' 2 cos A' 

The next step in our transformation consists in finding con- 
venient and at the same time sufficiently accurate expressions 
of aA and aIT. Let 

r, R= tho true refractions for the apparent altitades A' and 

then we have, within less than 0".l, 

aA = TTi cos (A' — r) — r 


If we neglect r in the term tt^ cos (A' — r), the error in this term 
will never exceed 1" ; but even this error will be avoided by 
taking the approximate expression 

cos (A' — r) = cos A' + sin r sin A' 

and we shall then have 

aA = TTi cos A' — r -\- TTi sin r sin A' 

/ If n/i . ^1 ^^^ r 8inA'\ 
= (ri cos A'— r) 1 H I 

\ TTi COS A' — r / 

Since the second term of the second factor produces but 1" 

in aA, we may employ for it an approximate value, which will 

still give aA with great precision. Denoting this term by A, we 


Tf, sin r sin A' sin r tan A' 

k = 

^1 cos A' — r I 

^1 cos A' 

or, very nearly, 

/: = sin r tan A' 1 1 -J \ 

\ TTi cos A' / 

If we put 

r = a cot A', 

in which a has the value given in Table 11., we have 

A:=asinr(l + — ^\ 
\ TTi Sin A / 

Now, a increases with A', but in such a ratio that k remains very 
nearly constant for a constant value of ;rp We may without 
sensible error take tt^ = 57' 30" = 3450", which is about the 
mean value of ;ri, and we shall find for a mean state of the air, 
by the values of a given in Table IL, 

forA'= 5° A =z= 0.000291 

A' =45 A = 0.000286 

A'= 90 k = 0.000285 

Hence, if we take 

k = 0.00029 
the formula 

A A = (tti cos A' — r) (1 + k) (452) 

will give aA within ^i^Anny ^^ ^^ whole amount, that is, within less 
than 0".02 in a mean state of the air. For extreme variations 


of the density of the air, it is possible that the refraction may 

bo increased by its one-sixth part, and k will also be increased 

by its one-sixth part. But, as the term depending ou k is not 

more than 1", the error in aA, even in the improbable case 

supposed, will not be greater than 0'M6. The formula (452) 

may therefore be regarded as practically exact with the value 

k = 0.00029. 

A strict computation of the sun's or a planet's altitude requires 

the formula 

A jy = 22 — P cos (J' — R) 

but P is in all cases so small that the formula 

£^H = R—Pco^H' (453) 

will always be correct within a very small fraction of a second. 
Now, let 

cos A' cos J?' ' 

The quantities r' and W computed from the mean values of the 
refraction are given in Table XTV. under the name "Mean 
Reduced Refraction for Lunars." The numbers of the table 
are corrected for the height of the barometer and thermometer 
by means of Table XIV.A and B. These tables are computed 
from Bessel*s refraction table, assuming the attached ther- 
mometer of the barometer, and the external thermometer, to 
indicate the same temperature, which is allowable in our present 
problem.* By the introduction of r' and iZ', we obtain 

£^=(-.-0(1 + A) 7^Jlr--R'-P 

cos A cos M 

and the coefficients of formula {g) become 

* If it is desired to compute r^ and R' with the utmost rigor, it can be done bj 
Table II., by taking (Art. 107) 

sin A' sin If' 

The tables XTV. and XIV.A and B gire the correct Talues to the nearest second in all 
practical cases. 


Ai = (t:, — r') (1 + A:) sin (A' + i aA) 

^ ^ V T y 2 COS J?' 

Ci = — (i?' — P) sin (J?'— i AlT) 

^ 2 cos A' 

The term A^ Q sin 1" cos rf' is very small, its maximnm being 
only 1". It is easy to obtain an approximate expression for it 
and to combine it with the term -4iC0srf'. In so small a term 
we may take 

Cy sin 1"= — JZ'sin T'sin ^=— sin R tan S'= — A- 

and hence 

A, — AlCi sin 1"= ill (1 + A) = (r^ — r') (1 + A)« sin (A' + } aA) 

If now we put 

A = (1 + *)' . ?^l^i*l±i^ 

sin h' 

sin 2Sr 

^_ 8in(g^— JAJT) 
"" sin^ 

_ sin (2 h' + aA) 
"" sin 2 A' 

^' = (n^ —r^A sin A' cot d' 
^ = — (r, — r') J9 sin J' cosoc d' 
C"= — {B'—P) C sin jy' cot d' 
D'= (i?'— P) Dsin A'cosec d' 

the formula (g) becomes, when divided by sin rf', 




sin d' 
the first member of which may be put under the form 

,/, . 2 sin iA</ cos (</'+ iA<f) \ 

A^l 1 + . ' - 

\ sm d' I 


80 that if we put 

_ AcP sin r^ cos (d' + i^d) 

^ "~ 2 Bin d' 

or, within 0'M5, 

ar = — J A<f« sin 1" cot d' (457) 

we have 

Ad = A'+B'+ C" + D' + X (468) 

The terms A'y B'^ C", and D' are computed directly from the 
apparent distance and altitudes by (456), and with sufficient 
accuracy with four-figure logarithms. The logarithms of ^, jB, C^ D, 
are given in Table XV., log A and log D with the arguments 
n^ — r' and A'; log B and log C with the arguments J?' — P 
and -H'. In the construction of this table a/i and a-BT are com- 
puted by (452) and (453), and then the logarithms of Aj By Q D, 
by (455). 

The sum A'+ B'+C' + D' is called the "first correction of the 
distance," and, being very nearly equal to Arf, is used as the argu- 
ment of Table XVI., which gives x, or the " second correction 
of the distance," computed by (457). When x is greater than 80" 
and the distance small, it will be necessary to enter this table a 
second time with the more correct value of Ad found by em- 
ploying the first value of x. 

The correction a^ being thus found and added to rf', we have 
rfp or the distance reduced to the point 0. The reduction to the 
centre of the earth is then made by (450). This reduction is 
also facilitated by a table. If we put 

and then 

,_ . / sin J sin d \ 

N= AttI I 

\ sin d^ tan <f, / 

. sin ^ - . sin J 

a = — A^ = At: 

tan d^ sin d^ 

we shall have 

N=a + b (459) 

and a and b can be taken from Table XIX. where a is called " the 
first part of iV^," and b " the second part of N.** We then have 

d — ^, = iVsin^ (4aO) 

which is the correction to be added to rfj to obtain the geocentric 
distance d. Table XIX. is computed with the mean value of 


IT = 57' 30", which will not produce more than 1" error in 
d — rfj in any case. But, if we wish to compute the correction 
for the actual parallax, we shall have, after finding N by the 

d-.ef,= iVrflin^X^ (460*) 

TT being in seconds. 

The trouble of finding the declinations of the bodies and the 
HBe of Table XIX. would be saved if the Almanac contained the 
logarithm of N in connection with the lunar Ephemeris. The 
value of log iVin the Almanac would, of course, be computed 
with the actual parallax, and (460) would be perfectly exact. 

We have yet to introduce corrections for the elliptical figure 
of the discs of the moon and sun produced by refraction. These 
corrections are obtained by Tables XVII. and XVJLLl., which aro 
constructed upon the following principles. Let 

A5j, aS^= tho contractions of the vertical semidiamoters, 
^s, aS = the contractions of the inclined semidiametors; 

then we have (Art. 133) 

A5 = A«i cos' q aS= Afif, cos' Q 

where j = the angle ZM'S' (Fig. 29) and Q = ZS'M'. We 

sin H' — sin A' cos d' 

cos q = 

cos A' sin d' 

But, by (456), 

sin ir J5' sin A' cos rf' A' 

cos A' sind' B (rj — r') cos A' cos A' sin d' A (r^ — r') cos A' 

so that 

cos q = — I 1 

^ \ ^ ^ J9 / (;r^ - r') cos A' 

If we put A = 1 and jB = 1, which are approximate values, we 

shall have 

A'+ B' 

cos q = -7 

^ (t,— OcosA' 

^s = ^s, r ^'+^' T (461) 

^L(;rj--r')C08A'J ^ ^ 


In order to ascertain the degree of accuracy of thb formula, 
we observe that the errors in cos q produced by the assumption 
A = lj B = ly are 

.. -.tan A' Bin IT 

e = (A — l) - — J c' = (1 — jB) 

tand' cos h* Bind' 

the errors in cos' q are 

2ecosjf 2€^cosq 

and the errors in a 5 are, therefore, 

___ 2as, (A — 1) tan h' cos q , 2as, (1 — ^) sin JT cos ^ 

* tan d' ^ cos A' sin d' 

In order to represent extreme cases, let us suppose j = and 
S^= 90®, which will give e^ and e/ their greatest values; then 
we shall find for the diflTerent values of A' the following errors : 


e, Un d' 

e,' sin d' 
















it can only be for very small values of rf' that the error e^ can be 
important, even for A' = 5^ ; and, as these small values of the 
distance are always avoided in practice, our formula (461) may 
be considered quite perfect. 

In the same manner, we shall find 

» l(^E' — P) cos j&r'J ^ ^ 

which is even more accurate than (461). 

These formulae are put into tables as follows. For the moon, 
Table XVII.A, with the arguments A' and;:^— r', gives the 
value of 

9 = X/ 

^ (r^ — O* C08» A' -^ 

where /is an arbitrary factor (=18000000) employed to give^ 
convenient integral vahies. Then Table X^TI.B, with the argu- 
ments^ and A'+ £\ gives 


For the sun, Table XVm. A, with the arguments jff' and JR ' — P, 
gives the value of 


in which F= -^; and Table XVm.B gives 


In these tables A' + B' is called the " whole correction of the 
moon," and C' + D' the "whole correction of the sun/' As 
these quantities are furnished by the previous computation of 
the true distance, the required corrections are taken from the 
tables without any additional computation. 

The values of a5 and ^S are applied to the distance as follows : 
when the limb of the moon nearest to the star or planet is 
observed, a5 is to be subtracted, and when the farthest limb is 
observed, a5 is to be added ; when the sun is observed, both ^s 
and AiS are to be subtracted from d. 

In strictness, these corrections should be applied to the dis- 
tance rf', and the distance thus corrected should be employed in 
computing the values of A\ B\ C", and D'. This would 
require a repetition of the computation after a5 and ^S had been 
found by a first computation; but this repetition will rarely 
change the result by 0".5. In the extreme and improbable case 
when the distance is only 20° and one body is at the altitude 5® 
and the other directly above it in the same vertical circle (so that 
the entire contraction of the vertical semidiameter comes into 
account), such a repetition would change the result only 1".8 ; 
and even this error is much less than the probable error of 
sextant observations at this small altitude, where the sun and 
moon already cease to present perfectly defined discs. 

250. I shall now recapitulate the steps of this method. 
1st. The local mean time of the observation being T^ and the 
assumed longitude i, take from the Ephemeris, for the approxi- 



mate Greenwich time T+ L^ the quantities 5, Sj ff, P, d, and J. 
(For the sun we may always take P = 8".5 ; for a star, iS ~ 0, 

2d. If A'', -ff", d" denote the observed altitudes and distance 
of the limbs, find 

s'= 8 -\- correction of Table XIL, 
7rj== 7:+ correction of Table XIIL, 

and the apparent altitudes and distance of the centres, 


W=H"^ Sy 

d'=d"±«'dz S 

upper signs for upper and nearest limbs, lower signs for lower 
and farthest limbs. 

For the altitudes h' and H'^ take the " reduced refractions" 
r' and R from Table XTV., correcting them by Table XIV.A 
and B for the barometer and thermometer. Then compute the 

A' = (tr,— r')^8inA'cotd' C" = — (i?'— P)C8inJrcotrf' 

B'=^ (r, — r') -B sin W cosec d' D'= (^R'— P)JD sink' cosecd' 

for which the logarithms of Ay P, Q and D are taken from 
Table XV. In this table the argument n^ — r' is called the 
" reduced parallax and refraction of the moon," and P' — Pthe 
" reduced refraction and parallax of the sun (or planet) or star." 
For a star this argument is simply P'. 

When rf'> 90°, the signs of A' and C will be reversed. It 
may be convenient for the computer to determine the signs by 
referring to the following table : 

d' < 90° 
d' > 90° 









3d. The terms A* and P', which depend upon the moon's 
parallax and refraction, may be called the first and second parts 
of the moon's correction, and the sum A* + P' the " whole cor- 
rection of the moon." In like manner, C" and P'may be called 
the first and second parts of the sun's, planet's, or star's correc- 


tioii, and the sura C -|- ly the " whole correction of the sun, 
planet, or star." 

The sum of these corrections = ^' + ^' + C + D' may be 
called the "first correction of the distance.** Taking it as the 
upper argument in Table XVI., find the second correction = a:, 
the sign of which is indicated in the table. 

4th. Take from Table XVII.A and B the contraction of its 
inclined semidiameter = a5. If the sun is the other body, take 
also the contraction from Table A V 111. A and B, = a5. The 
sign of either of these corrections will be positive when the 
farthest limb is observed, and negative when the nearest limb is 

5th. The correction for the compression of the earth is = 
J\r sin ifyif being the latitude ; and N may be accurately com- 
puted by the formula 

\ sin dy^ tan d^^ I 

or it may be found within 1" by Table XIX., the mode of con- 
sulting which is e\ndent. The sign of iVsin ^ will be determined 
by the signs of iVand sin ^, remembering that for south latitudes 
sin f is negative. 

All the corrections being applied to rf', we have the geocen- 
tric distance d; and hence the corresponding Greenwich time 
and the longitude. 

Example. — ^Let us take the example of the preceding article 
(p. 399), in which the observation gives 

1866, March 9th, ^ = 85<>. 

T = 5* 14" 6* 2 ^" = 52^ 34' 0" Barom. 29.5 in. 
Assumed i = 10 O -ff"= 8 66 23 Thorm.58^F. 

Approx. Gr. T. = 15 14 6 B G d" =44 36 58.6 
By the Ephemeris, we have 

s = 16' 23".l r = 60' 1".9 S= 16' 8".0 P= 8".6 

Table XII. +14 .0 Tab. XIII. + 3 .9 d = + U^ J= — 4« 

«'=1637.1 r,= 605.8 

The computation may be arranged as follows: 




620 84^,0 


80 66'.4 


»' = 

+ 16.6 

s = 


• = 16 87 .1 

A' = 

62 60.6 

J7' = 

9 12.6 

5= 16 8.0 
d'^45 9 « .7 

Table XIV. 
" " A. 
** " B. 

1 . 
— 1 . 

P — 

— 6 . 
6 . 

Ill .1 
60 6 .8 

6 87 .6 
8 .6 


68 64 .7 

J?' P — 

6 29 .0 

(Table XV.) log ^ 0.0019 
log (ttj — r') 8.6484 
log sin h' 9.9016 
log cot d' 9.9976 

(Table XV.) log C 9.9978 
log {If— P) 2.6172 
log sin ZT' 9.2042 
log cot d' 9.9976 


A' — 

+ 46'63".9 



(Table XV.) log B 9.9981 

log (tj — r') 8.6484 

log sin W 9.2042 

log cosec d' 0.1498 

log ^' 

-»' = — 13'14".8 
-4'+ ^'=-1-83 89 .6 

Table XIX. Ist Part of iV= — 6" 
2d •* " = — 2 

(Table XV.) log Z> 9.9987 

log(iZ'— P) 2.6172 

log sin h' 9.9016 

log ooseo d' 0.1498 

log ly 2.6667 

lynz-f 6' 8".7 
C'+i>'=-f-6 16 .6 Istcorr. = +88'66''J 

(Table XVI.) 2d corr. = — 18 .5 

(Table XVII.) A* = 0. 

(Table XVIII.) a5= — 9 . 

8^0. iV sin ^ = — 4.6 

</ = 46 48 12 .8 

This result agrees with that found by the rigorous method on 
p. 401, within 1". 

To find the longitude, we now have, by the American Ephe- 
meris for March 9, 

(T) = 15* 0* 0* (O = 46° 4^ 64" Q = 0.2510 JQ= + n 

d = 45 48 13 

7 19 log = 2.6425 
log t = 2.8935 

t = 



Table XX. 






T — 




L — 





251. In consequence of the neglect of the fractions of a second 
in several parts of the above method, it is possible that the computed 
distance may be in error several seconds, but it is easily seen 
that the error from this cause will be most sensible in cases 
where the distance is small ; and, since the lunar distances are 
given in the Ephemeris for a number of objects, the observer 
can rarely be obliged to employ a small distance. If he confines 
himself to distances greater than 46® (as he may readily do), the 
method will rarely be in error so much as 2", especially if ho 
also avoids altitudes less than 10*^. Wlien we remember that 
the least count of the sextant reading is 10'', and that to the 
probable error of observation we must add the errors of gradua- 
tion, of eccentricity, and of the index correction, it must be con- 
ceded that we cannot hope to reduce the probable error of an 
observed distance below 6", if indeed we can reduce it below 
10". Our approximate method is, therefore, for all practical 
purposes, a perfect method, in relation to our present means of 

252. K the altitudes have not been observed, they may be 
computed from the hour angles and declinations of the bodies, 
the hour angles being found from the local time and the right 
ascensions. But the declination and right ascension of the moon 
will be taken from the Ephemeris for the approximate Green- 
wich time found wdth the assumed longitude. If, then, the assumed 
longitude is greatly in error, a repetition of the computation may 
be necessary, starting from the Greenwich time furnished by the 
first. As a practical rule, we may be satisfied with the first 
computation when the error in the assumed longitude is not 
more than 30*. In the determination of the longitude of a fixed 
point on land, it will be advisable to omit the observation of the 
altitudes, as thereby the observer gains time to multiply the 
observations of the distance. But at sea, where an immediate 
result is required with the least expenditure of figures, the alti- 
tudes should be observed. 

253. At sea, the observation is noted by a chronometer regu- 
lated to Greenwich time, and the most direct employment of the 
resulting Greenwich time will then be to determine the true 
correction of the chronometer. This proceeding has the advan- 


tage of not requiring an exact determination of the local time at 
the instant of the observation. 

For example, suppose the observation in the example above 
computed had been noted by a Greenwich mean time chrono- 
meter which gave 15* 10"* ()•, and was supposed to be slow 4" 6*. 
The true Greenwich time according to the lunar observation 
was 15* 13"* 0*, and hence the true correction was + S" 0*. With 
this correction we may at any convenient time afterwards deter- 
mine the longitude by the chronometer (Art. 214). 

In this way the navigator may from time to time during a 
voyage determine the correction of the chronometer, and, by 
taking the mean of all his results, obtain a very reliable correc- 
tion to be used when approaching the land. He may even 
determine the rate of the chronometer with considerable acca- 
racy by comparing the mean of a number of observations in 
the first part of the voyage with a similar mean in the latter 
part of it. 

254. To correct the longitude found by a lunar distance for errors 
of the Ephemeris. — In relation to the degree of accuracy of the 
observation, we may in the present state of the Ephemeris regard 
all its errors as insensible except those which affect the moon*8 
place. If, therefore, the longitude of a fixed point has been 
found by a lunar distance on a certain date, the corrections of 
the moon's right ascension and declination are first to be found 
for that date from the observations at one or more of the prin- 
cipal obscrv^atories, and then the correction of the longitude ynXl 
be found as follows. Let 

o, o =: the right ascension and declination of the moon given 

in the Ephemeris for the date of the ohservation, 
A, il = those of the sun, planet, or star, 
dafdd=^ the corrections of the moon's right ascension and 
(id = the corresponding correction of the lunar distance, 
SL — - the corresponding correction of the computed longi- 

In Fig. 30, M and S being the geocentric places of the two 
bodies, as given in the Ephemeris, and d denoting the distance 
MS, we have 

cos d = Hin d sin J -f cos (J cos J cos (a — - A) (463) 


by diflferentiating which we find 

- , cos d cos J sin (a — A) . 

da = ^^ i . oa 

sin d 

COS ^ sin J — sin ^ cos J cos (a — A) .^ ,^^, 

: — ^^ ^ . M (464) 

sin a ^ ' 

If then 

V = the change of distance in 3*^ 

we shall have 

dL = — ddX- (465) 

in computing which we employ the proportional logarithm of the 
Ephemeris, Q = log — , reduced to the time of the observation. 

Example. — At the time of the observation computed in Art. 
250, we have 

Moon, a = 2* 11* 14* ^ = + 14^ 18'.4 

Sun, ^ = 23 22 25 j=— 4 3.1 

a — ^= 2 49 19 d= 45 48.2 

= 42«19'.8 

with which we find, by (464), 

dd = 0.908 ^o + 0.350 M 

and hence, by (465), with log Q = 0.2511, 

^i = — 1.62 ^o — 0.62 dd 

Suppose then we find from the Greenwich observations da = 
— O-.SS = — 5".7 and dd = — 4''.0, the correction of the longi- 
tude above found will be 

255. To find the longitude by a lunar distance not given in the 
Ephemeris. — The regular lunar-distance stars mentioned in Art. 
247 are selected nearly in the moon's path, and are therefore in 
general most favorable for the accurate determination of the 
Greenwich time. Nevertheless, it may occasionally be found 
expedient to employ other stars, not too far from the ecliptic. 
Sometimes, too, a difl'erent star may have been observed by 
mistake, and it may be important to make use of the observation. 

Vol. L— 27 


The true distance d is to be found from the observed distance 
by the preceding methods, as in any other case. Let the local 
time of the observation be T, and the assumed longitude L. 
Take from the Ephemeris the moon's right ascension a and de- 
clination d for the Greenwich time T + L^ and also the star g 
right ascension A and declination J ; with which the corresp<Mid- 
ing true distance dgis found by the formula 

cos d^ = sin ^ sin J -{■ cos d cos A cos (a — A) 

Then, if d = d^j the assumed longitude is correct ; if otherwise, 


;i = the increase of a in one minute of mean time, 

fi = the increase of ^ « " *^ " 

Y = the increase of d " " '* " 

then we have, by (464), 

cos d cos J sin (o — A) , cos ^ sin J — sin d cos J cos (a — A) ^ 

y^^ 1 L,l ; i^ d.jj 

sin d^ sin d^ 

and hence the correction of the assumed longitude in seconds 
of time, 


For computation by logarithms, these formulse may be ar- 
ranged as follows : 

tan J 
tanJf = 

cos (o — A) 

sin J cos (d — IdTs 

coBd,= :-^— 

sm Jf 

, cos ^ cos J sin (a — -4) , _ ... .. .-.. 



r i 

Example. — Suppose an observer has measured the distance 
of the moon from Arciurus, at the local mean time 1856 March 
16, r = 10* 30'" 0*, in the assumed longitude i = 6* O* 0*, and, 
reducing his observ^ation, finds the true distance 

d = 73« 55' 10" 
what is the true longitude ? 


For the Greemvich time T+ L = 16^ 30"» we find 

a = 8* 47- 6-.54 ^ = + 23° 12' 7'M ^ = + 31".40 

A= 14 9 7 .04 J = + 19 55 44 .8 /9= — 8 .62 

a—A=— 5* 22" 0-.50 = — 80° 30' 7".5 

with which we find by (466), 

d^ = 73° 55' 35". r = — 25".59 

d—d^ = — 25" dL = + 58- .6 

and therefore the longitude is 6* 0~ 58*.6. 

256. In order to eliminate as far as possible any constant 
errors of the instrument used in measuring the distance, wo 
should observe distances from stars both east and west of the 
moon. K the index correction of the sextant is in error, the 
errors produced in the computed Greenwich time, and conse- 
quently in the longitude, will have different signs for the two 
observations, and will be very nearly equal numerically: they will 
therefore be nearly eliminated in the mean. K, moreover, the 
distances are nearly equal, the eccentricity of the sextant will 
have nearly the same effect upon each distance, and will there- 
fore be eliminated at the same time with the index error. Since 
even the best sextants are liable to an error of eccentricity of as 
much as 20", according to the confession of the most skilful 
makers, and this error is not readily determined, it is important 
to eliminate it in this manner whenever practicable. K a circle 
of reflexion is employed which is read off by two opposite 
verniers, the eccentricity is eliminated from each observation ; 
but even with such an instrument the same method of observa- 
tion should be followed, in order to eliminate other constant 

It has been stated by some writers that by observing distances 
of stars on opposite sides of the moon we also eliminate a con- 
stant error of observatiorij such, for example, as arises from a 
faulty habit of the observer in making the contact of the moon's 
limb with the star. This, however, is a mistake; for if the 
habit of the observer is to make the contact too closCy that is, to 
bring the reflected image of the moon's limb somewhat over 
the star, the effect will be to increase a distance on one side of 
the moon while it diminishes that on the opposite side, and the 
effect upon the deduced Greenwich time will be the same in 


both cases. This will be evident from the following diagram^ 

(Fig. 31). Suppose a and b 
^*^\f^- are the two stars, M the 

moon's limb. If the observer 
* [j • judges a contact to exist when 

the star appears within the 
moon's disc as at (?, the distance 
ac is too small and the distance 
be too great. But, supposing the moon to be moving in the direc- 
tion from a to 6, each distance will give too early a Greenwich 
time, for each will give the time when the moon's limb was 
actually at c. 

If, however, we observe the sini in both positions, this kind 
of error, if really constant, will be eliminated ; for, the moon's 
bright limb being always turned towards the sun, the error will 
increase both distances, and \vill produce errors of opposite sign 
in the Greenwich time. Hence, if a series of lunar distances 
from the sun has been observed, it wnll be advisable to form two 
distinct means, — one, of all the results obtained from increasing 
distances, the other, of all those obtained from decreasing dis- 
tances : the mean of these means ^vill be nearly or quite free 
from a constant error of observation, and also from constant in- 
strumental errors. 


257. By chronometers, — This method is now in almost universial 
use. The form under which it is applied at sea differs verj- 
slightly from that given in Art. 214. The correction of the 
chronometer on the time of the first meridian (that of Green- 
\^'ich among Aniorieau and English navigators) is found at any 
place whose longitude is known, and at the same time also it<s 
daily rate is to be established with all possible care. The rate 
being duly allowed for from day to day during the voyage, the 
Greenwich time is constantly known, and therefore at anv 
instant when the local time is obtained bv obser\'ation, the Ion- 
gitude of the ship is determined. 

The local time on shipboard is always found from an altitude 
of some celestial object, observed with the sextant from the sea 
horizon. (Art. 156.) The computation of the hour angle is 
then made by (208), and the resulting local time is comjiaretl 
directly with the Greenwich time given by the chronometer at 



the instant of the observation. The data from the Ephemeris 
required in computing the local time are taken for the Green^nch 
time given by the chronometer. 

Example. — A ship being about to sail from New York, th<!: 
master determined the correction on Greenwich time and the 
rate of his chronometer by observations on two dates, as follows: 

1860 April 22, at Greenwich noon, chron. correction = -f S" lO'.O 
** " 30, " " " " = + 3 43 .6 

Kate in 8 days = + 33 .6 
Daily rate = + 4 .2 

On May 18 following, about 7* 30* A.M., the ship being in lati- 
tude 41° 88' N., three altitudes of the sun's lower limb were 
obsen'ed from the sea horizon as below. The correction of the 
chronometer on that day is found from the correction on April 30 
by adding the rate for 18 days. (It will not usually be worth 
while to regard the fraction of a day in computing the total rate 
at sea.) The record of the observation and the whole computa- 
tion may be arranged as follows : 

O 290 40/ 10// Barom. 80.32 *». 

•* 46 Therm. 69« F. 

«« 60 60 

Mean = 29 46 40 

Index corr. = — 1 10 

Dip = — 4 2 

29 40 28 

Semid. = + 16 60 

Refraction = — 1 42 

Parallax = -f- 8 

A = 29 64 44 

^ = 41 SS 

P = 70 21 21 

* = 70 64 88 

, _ A = 40 69 49 

1860 Maj 18. 

^ = 

: 41<» 88' 


9* 87"« 21'. 

" 87 63. 
*• 88 20. 


— 9 37 61.8 


= -f- 4 69.2 

Gr. date = Maj 17, 21 42 60.6 

for which time we take from the 
Ephemeris the quantities 

0'8 S = 19® 88' 89" 
Semidiameter = 16' 60" 

Equation of time = — 3" 49'.8 

sec 0.12688 
cosec 0.02604 

cos 9.61464 
•in _9. 81 092 


sin 9.74174 

Apparent time = 7*32-* 6».3 
Eq. of time = — 3 49.8 

Local mean time = 19 28 16.6 
Gr. ** " r= 21 42 60.6 

Longitude = 2 14 34 = 38« 38'.6 W. 

In this observation, the sun was near the prime vertical, a posi- 
tion most favorable to accuracy (Art. 149). 


The method by equal altitudes may also be used for finding 
the time at sea in low latitudes, as in Arts. 158, 159. 

258. In order that the longitude thus found shall be worthy 
of confidence, the greatest care must be bestowed upon the 
determination of the rate. As a single chronometer might 
deviate very greatly without being distrusted by the navigator, 
it is well to have at least three chronometers, and to take the 
mean of the longitudes which they severally give in every case. 

But, whatever care may have been taken in determining the 
rate on shore, the sea rate will generally be found to differ from 
it more or less, as the instrument is affected by the motion of the 
ship ; and, since a cause which accelerates or retards one chro- 
nometer may produce the same effect upon the others, the agree- 
ment of even three chronometers is not an absolutely certain 
proof of their correctness. The sea rate may be found by 
determining the chronometer correction at t>vo ports whose 
difference of longitude is well known, although the absolute 
longitudes of both ports may be somewhat uncertain. For thli 
pui-pose, a " Table of Chronometric Differences of Longitude*' is 
given in Raper*s Practice of Navigaiiorij the use of which U 
illustrated in the following example. 

Example. — At St. Ilelena, May 2, the correction of a chro- 
nometer on the local time was — 0* 23'" 10*.3. At the Cape of 
Good Hope, May 17, the correction on the local time was 
+ 1* 14*^ 28*. 6 ; what was the sea rate ? 

We have 

Corr. at St. Ilelena, May 2d = — 0* 23- IC.S 

Chron. diff. of long, from Raper = + 1 36 45. 

Corr. for Capo of G. H., May 2d =+ 1 13 34.7 
*< " *' " 17th = + 1 14 28 .6 

Rate in 15 days = -{- 53 .9 

Daily Hoa rate = + 3 .59 

259. By lunar distances. — Chronometers, however perfectly 
made, are liable to derangement, and cannot be implicitly relied 
upon in a long voyage. The method of lunar distances (Art^ 
247-256) is, therefore, employed as an occasional check upon the 
chronometers even where the latter are used for finding the 
longitude from day to day. When there is no chronometer on 

AT 6£A. 423 

board, the method of lunar distances is the only regularly avail- 
able method for finding the longitude at sea, at once sufficiently 
accurate and sufficiently simple. 

As a check upon the chronometer, the result of a lunar distance 
is used as in Art. 253. 

In long voyages an assiduous observer may determine the sea 
rates of his chronometers with considerable precision. For this 
purpose, it is expedient to combine observations taken at various 
times during a lunation in such a manner as to eliminate as far 
as possible constant errors of the sextant and of the obser\'er (Art. 
256). Suppose distances of the sun are employed exclusively. 
Let two chronometer corrections be found from two nearly equal 
distances measured on opposite sides of the sun on two different 
dates, in the first and second half of the lunation respectively. 
The mean of these corrections will be the correction for the 
mean date, very nearly free from constant instrumental and 
personal errors. Li like manner, any number of pairs of equal, 
or nearly equal, distances may be combined, and a mean chro- 
nometer correction determined for a mean date from all the 
observations of the lunation. The sea rate will be found by 
comparing two corrections thus determined in two difterent 
lunations. This method has been successfully applied in voyages 
between England and India. 

260. By the eclipses of Jupiter's satellites, — An observed eclipse 
of one of Jupiter's satellites furnishes immediately the Green- 
wich time without any computation (Art. 225.) But the eclipse 
is not sufficiently instantaneous to give great accuracy ; for, with 
the ordinary spy-glass with which the eclipse may be observed 
on board ship, the time of the disappearance of the satellite may 
precede the true time of total eclipse by even a whole minute. 
The time of disappearance will also vary with the clearness of 
the atmosphere. Since, however, the same causes which accele- 
rate the disappearance will retard the reappearance, if both 
phenomena are observed on the same evening under nearly the 
same atmospheric conditions, the mean of the two resulting 
longitudes will be nearly correct. Still, the method has not the 
advantage possessed by lunar distances of being almost always 
available at times suited to the convenience of the na\dgator. 

261. By the moon's altitude. — This method, as given in Art. 243, 


may be used at sea in low latitudes ; but, on account of tbe 
unavoidable inaccuracy of an altitude observed from the set 
horizon, it is even less accurate than the method of the preceding 
article, and always far inferior to the method of lunar distances, 
although on shore it is one which admits of a high degree of 
precision when carried out as in Art. 245. 

262. By occultaiions of stars by the moon. — This method, which 
will be treated of in the chapter on eclipses, may be successfiillj 
used at sea, as the disappearance of a star behind the moon*t 
limb may be observed with a common spy-glass at sea with 
nearly as great a degree of precision as on shore ; but, on account 
of the length of the preliminary computations as well bb of the 
subsequent reduction of the observation, it is seldom that a 
navigator would think of resorting to it as a substitute for the 
convenient method of lunar distances. 



263. In the preceding two chapters we have treated of 
methods of finding the position of a point on the earth's surface 
by thfi two co-ordinates latitude and longitude; and therefore in all 
these methods the required position is determined by the inter- 
section of two circles, one a parallel of latitude and the other a 
meridian. In the following method it is determined by circles 
oblique to the parallels of latitude and the meridians. The prin- 
ciple which underlies the method has often been applied ; but its 
value as a practical nautical method was first clearly shown by 
Capt. Thomas II. Sumner.* 

Let an altitude of the sun (or any other object) be observed 
at any time, the time being noted by a chronometer regulated to 
Greenwich time. Suppose that at this Greenwich time the sun 

♦ A new and accurate method of finding a thip's position at tea by projection (m 
tor's chart: by Capt. Thomas H. Sumneb. Boston, 1843. 

Sumner's method. 425 

is vertical to an observer at the point M of the globe (Fig 32). 
Let a small circle A A' A" be described on 
the globe from Jif as a pole, with a polar dis- Fig^2. 

tance MA equal to the zenith distance, or 
complement of the observed altitude, of the 
Bun. It is evident that at all places within 
this circle an observer would at the given 
time observe a smaller zenith distance, and 
at all places without this circle a greater 
zenith distance; and therefore the observa- 
tion fully determines the observer to be on 
the circumference of the small circle AA'A'^. If, then, the 
navigator can project this small circle upon an artificial globe or 
a chart, the knowledge that he is upon ihi$ circle will bejitst as valuable 
to him in enabling him to avoid dangers as the knowledge of either his 
latitude alone or his longitude alone; since one of the latter elements 
only determines a point to be in a certain circle, without fixing 
upon any particular point of that circle. 

The small circle of the globe described from the projection of 
the celestial object as a pole we shall call a circle of position. 

264. To find the place on the globe at which the sun is vertical {or the 
sun's projection on the globe) at a given Ghreenwich time, — The sun's 
hour angle from the Greenwich meridian is the Greenwich 
apparent time. The diurnal motion of the earth brings the sun 
into the zenith of all the places whose latitude is just equal to 
the sun's declination. Hence the required projection of the 
Bun is a place whose longitude (reckoned westward from Green- 
wich from 0* to 24*) is equal to the Greenwich apparent time, 
and whose latitude is equal to the sun's declination at that time. 

265. From an altitude of the sun taken at a given Greenwich time, 
to find the circle of position of the observer , by projection on an artificial 
globe. — ^Find the Greenwich apparent time and the sun's declina- 
tion, and put down on the globe the sun's projection by the 
preceding article. From this point as a pole, describe a small 
circle with a circular radius equal to the true zenith distance 
deduced from the observation. This will be the required circle 
of position. 

266. Tlie preceding problem may be extended to any celestial 


object. The pole of the circle of position will always be th^ 
place whose west longitude is the Greenwich hour angle of the 
object (reckoned from 0* to 24*) and whose latitude is the decli- 
nation of the object. The hour angle is found by Art. 64. 

267. To find both the laiiiude and the longitude of a ship by circles of 
position projected on an artificial globe. — First. Take the altitudes 
of two different objects at the same time by the Greenwich 
chronometer. Put down on the globe, by the preceding problem, 
their two circles of position. The observer, being in the ciream- 
fcrence of each of these circles, must be at one of their two points 
of intersection ; which of the two, he can generally determine 
from an approximate knowledge of his position. 

Second. Let the same object be observed at two different limefl^ 
and project a circle of position for each. Their intersection 
gives the j^osition of the ship as before. If between the observa- 
tions the ship has moved, the first altitude must be reduced to 
the second place of observation by applying the correction of 
Art. 209, formula (380). The projection then gives the ships 
position at the second observation. 

268. From an altitude of a celestial body taken at a given GfreenKick 
time, to find the circle of position of the observer , by projection on a 
Mcrcaior chart. — The scale upon which the largest artificial globes 
are constructed is much smaller than that of the working charts 
used by navigators. But on the Mercator chart a circle of 

position will be distorted, and can only 
Fig. 33. be laid down by points. Let i, i', L" 

(Fig. 33) be any parallels of latitude 
crossed by the required circle. For eaoli 
of these latitudes, with the true altitude 
found from the observation and the polar 
distance of the celestial body taken for 
the Greenwich time, compute the local 
time, and hence the longitude, '* by chro- 
nometer'' (Art. 257). Let /, /', /" be the 
longitudes thus found. Let A, A', A'' be the points whose 
latitudes and longitudes are, respectively, i, I; L\ V ; L'\ /"; 
these are evidently points of the required circle. The ship is 
consequently in the curve AA'A^', traced through these 


In practice it is generally sufficient to lay down only twc 
points ; for, the approximate position of tlie ship being known, 
if L and U are two latitudes between which the ship may be 
assumed to be, her position is known to be on the curve AA' 
somewhere between A and A'. When the difterence between 
L and U is small, the arc AA' will appear on the chart as a 
straight line. 

269. To find the latitude and longitude of a ship by circles of position 
projected on a Mercator chart. — IHrst. Let the altitudes of two 
objects be taken at the same time. Assume two latitudes em- 
bracing between them the ship's probable position, and find two 
points of each of their two circles of position by tlie preceding 
problem, and project these points on the chart. Each pair of 
points being joined by a straight line, 
tlie intersection of the two lines is 
very nearly the ship's position. Thus, 
if one object gives the points A^ A' 
(Fig. 34) corresponding to the lati- 
tudes i, Z/', and the other object the 
points B^ B' corresponding to the same latitudes, the ship's 
position is the point C, the intersection of AA' and BB'. 

It is, of course, not essential that the same latitudes should be 
used in computing the points of the two circles ; but it is more 
convenient, and saves some logarithms. 

If greater accuracy is desired, the circles may be more fully 
laid down by three or more points of each. 

Second. — The altitude of the same object may be taken at two 
different times, and the circles laid down as before ; the usual 
reduction of the first altitude being applied when the ship changes 
her position between the observations. 

It is evident from the nature of the above projection that the 
most favorable case for the accurate determination of the inter- 
section C is that in which the circles of position intersect at right 
angles. Hence the two objects observed, or the two positions 
of the same object, should, if possible, differ about 90° in azimuth. 
This agrees with the results of the analytical discussion of the 
method of finding the latitude by two altitudes. Art. 183. 

If the chronometer does not give the true Greenwich time, the 
only effect of the error will be to shift the point C towards the 
east or the west, without changing its latitude, unless the error U 


80 great as to affect sensibly the deelination which is taken from 
the Ephemeris for the time given by the chronometer. This method 
is, therefore, a convenient substitute for the usual method of find- 
ing the latitude at sea by two altitudes, a projection on the sailing 
chart being always sufiieient for the purposes of the navigator. 

Instead of reducing the fii'st altitude for the change of the ship's 
position between the observations, we may put down the circle 
of position for each observation and afterwards shift one of them 

by a quantity due to the ship's run. 

A' a' ^, Thus, let the first observation give the 

position line AA' (Fig. 35), and let Aa 
represent, in direction and length, the 
ship's course and distance sailed be- 
tween the observations. Draw aa! 
parallel to AA'. Then, BB' being the position line by the 
second observation, its intereection C vnXh aa' is the required 
position of the ship at the second observation. 

270. If the latitude is desired by computation, independently 
of the projection, it is readily found as follows. Let 

Zj, Zj = the longitudes (of A and B) found from the first and 
second altitudes respectively with the latitude X, 

Zj', Z/ = the longitudes (of A' and B!) found from the same 
altitudes with the latitude L\ 
Lq = the latitude of C. 

From Fig. 34 we have, by the similarity of the triangles ABC 

i;—i;i l^ — l,= B'C : BC 

(^l^^^l^')^(^l^-.l^') s l^ — U=BB' I BC = L'-^L I L^^L 

Uy^-j^n^^^-^ (467) 

(^/-O + C^i-'^ 



This formula reduces Sumner's method of " double altitudes** 
to that given long ago by Lalaxde (Astronomiey Art 3992, and 
Abrigi de Navigation, p. 68). The distinctive feature of Sumner*s 
proccfts, however, is that a single altitude taken at any time is 
made available for determining a line of the globe on which the 
ship is situated. 


271. To find the azimuth of the sun hy a position line projected on 
the chart. — ^Let AA' (Fig. 36) be a position line on 
the chart, derived from an observed altitude by 
Art. 268. At any point C of this line draw CM 
perpendicular to AA'^ and let NCS be the meri- 
dian passing through C; then SCM is evidently 
the sun's azimuth. The line CM is, of course, 
drawn on that side of the meridian N8 upon 
which the sun was known to be at the time of 
the observation. 

The solution is but approximate, since AA' should be a curve 
line, and the azimuth of the normal CM" would be diflFerent for 
different points of AA'. It is, however, quite accurate enough 
for the purpose of determining the variation of the compass at 
sea, which is the only practical application of this problem. 



272. The meridian line is the intersection of the plane of the 
meridian with the plane of the horizon. Some of the most use- 
ful methods of finding the direction of this line will here be 
briefly treated of; but the full discussion of the subject belongs 
to geodesy. 

273. By th£ meridian passage of a star. — If the precise instant 
when a star arrives at its greatest altitude could be accurately 
distmguished, the direction of the star at that instant, referred 
to the horizon, would give the direction of the meridian line ; but 
the altitude varies so slowly near the meridian that this method 
only serves to give a first approximation. 

274. By shadoics. — ^A good approximation may be made as 
follows. Plant a stake upon a level piece of ground, and give it 
a vertical position by means of a plumb line. Describe one or 

430 MERIDIAN line; 

more concentric circles on the ground from the foot of the stake 
as a centre. At the tw^o instants before and after noon when the 
shadow of the stake extends to the same circle, the azimuths of 
the shadow east and west are equal. The points of the circle at 
which the shadow terminates at these instants being marked, let 
the included arc be bisected ; the point of bisection and the centre 
of the stake then determine the meridian line. Theoretically, a 
small correction should be made for the sun's change of declina- 
tion, but it would be quite superfluous in this method. 

275. By single altitudes. — ^With an altitude and azimuth instru- 
ment, observe the altitude of a star at the instant of its passage 
over the middle vertical thread (at any time), and read the 
horizontal circle. Correct the observed altitude for refraction. 
Then, if 

h = the true altitude, 

<p = the latitude of the place of observation, 

jp = the star's polar distance, 

A = the star's azimuth, 

A' = the reading of the horizontal circle, 

we have, from the triangle formed by the zenith, the pole, and 
the star, 

tan. M = «i° (^ - f) «'» (^ - ^) (468) 

cos s cos (s — p) 

in which 

s = H9 + ^ + P) 

In this formula the latitude may be taken with the positive sign, 
whether north or south, and p is then to be reckoned from the 
elevated pole ; consequently, also, A will be the azimuth reckoned 
from the elevated pole. 

It is evident that in order to bring the telescope into the plane 
of the meridian we have only to revolve the instrument through 
the angle A, and therefore either J.' + J. or A' — A, according 
to the direction of the graduations of the circle, will be the 
reading of the horizontal circle when the telescope is in the 

The same method can be followed when the azimuth is ob- 
served with a compass and the altitude is measured \vith a sex- 
tant ; and then A' — A is the variation of the compass. 


276. From the first equation of (50), ip and 5 being constant, 
we have 

(f-4 = — 

cos A tan ^ 

and therefore an error in the obsei^ved altitude will have the 
least eftect upon the computed azimuth when tan y is a maxi- 
mum; that is, when the star is on the prime vertical. There- 
fore, in the practice of the preceding method the star should be 
as far from the meridian as possible. 

277. By equal altitudes of a star. — Observe the azimuth of a star 
with an altitude and azimuth instrument, or a compass, when at 
the same altitude east and west of the meridian. The mean of 
the tri'O readings of the instrument is the reading when its 
sight line is in the direction of the meridian. This is the 
method of Article 274, rendered accurate by the introduction 
of proper instruments for observing both the altitude and the 


. 278. If equal altitudes of the sun are employed, a correction 
for the change of the sun's declination is necessary, since equal 
azimuths will no longer correspond to equal altitudes. Let 

A' = the cast azimuth at the first observation^ 
il = " west " " second *^ 

d = the declination at noon, 
^d =: the increase of declination from the first to the second 

then, by (1), we have, A being the altitude in each case, 

sin (d — } A^) = sin f %mh — cos ip cos h cos A' 
sin (d + } A^) = sin ^ sin A — cos ^ cos A cos A 

the difference of which gives 

2 cos 5 sin } A^ = 2 cos <p cos A sin i (u4 + A') sin \{A — A') 

whence, since a5 is but a few minutes, we have, with sufficient 

A-A'= ^' ""^ ' (469) 

cos f cos A sin A 


It will be necessary to note the times of the two observations 
in order to find a5. If we take half the elapsed time as the 
hour angle t of the western observation, we shall have, instead 
of (469), the more convenient formula 

A-^' = ^^— (470) 

cos ^ sin t 

It will not be necessary to know the exact value of k, if only 
the same bistrumental altitude is employed at both observations. 

Now let -4/ and A^ be the readings of the horizontal circle at 

the two observations, then the readings corresponding to equal 

azimuths are 

a; and ^^ — (A — A') 

and, consequently, the reading for the meridian is the mean of 
these, or 

That is, the reading for the meridian is the mean of the ob- 
served readings diminished by one-half the correction (470), 
We here suppose the graduations to proceed from 0® to 860®, 
and from left to right. 

279. B\j the angular distance of the sun from any terrestrial object. — 
If the true azimuth of any object in view is known, the direction 
of the meridian is, of course, known also. The follo^ving method 
can be carried out with the sextant alone. Measure the angular 
distance of the sun's limb from any well-defined point of a 
distant terrestrial object, and note the time by a chronometer. 
Measure also the angular height of the terrestrial point above 
the horizontal plane. The correction of the chronometer being 
known, deduce the local apparent time, or the sun's hour angle t 
(Art. 54), and then with the sun's declination d and the latitude f 
compute the true altitude h and azimuth A of the sun by the 
formuhe (16), or 

,^ tan J ^ . tanfcos3f . , .. ,^ j /i-rtx 

tan Jf= , tan^= -— , tan A ^ cot (f — M)co8A(4il) 

cos ^ sin (95 — M) 

Now, let 0, Fig. 37, be the apparent position of the terrestrial 
point, projected upon the celestial sphere; Sihe apparent place 
of the sun, Z the zenith, P the pole ; and put 



D = the apparent angular distance of the ^ig. 8^ 

sun's centre from the terrestrial point 
= the observed distance increased by 

the sun's semidiameter, 
IIz= the apparent altitude of the point, 
h* = the sun's apparent altitude, 
a = the difference of the azimuth of the 

sun and the point, 
il' = the azimuth of the point. 

The apparent altitude A' will be deduced from the true altitude 
by adding the refraction and subtracting the parallax. Then in 
the triangle 8Z0 we have given the three sides ZS = 90° — A', 
ZO = 90° — H,80 = D, and hence the angle SZO = a can be 

found by the formula 

. ,. sin (5 — -ff) sin (5 — A') ....^ 

tan' } a = ^^ ^^ ^ (472) 

cos s cos (s — D) 

in which 

Then we have 

A' = A±a (473) 

and the proper sign of a to be used in this equation must be 
determined by the position of the sun with respect to the object 
at the time of the observation. 

K the altitude of the sun is observed, we can dispense with 
the computation of (471), and compute A by the formula (468). 
The chronometer Avill not then be required, but an approximate 
knowledge of the local time and the longitude is necessary in 
order to find d from the Ephemeris. 

If the terrestrial object is very remote, it will often suffice to 

regard its altitude as zero, and then we shall find that (472) 

reduces to 

tan i a = v^[tan i (D + N) tan J (-Z> — ^0] (474) 

This method is frequently used in hydrographic surveying to 
determine the meridian line of the chart. 

Example. — ^From a certain point 5 in a survey the azimuth 
of a point C is required from the following observation : 

Chronometer time =• 4* 12^ 12* 
Chronom. correction = — 2 

Local mean time = 4 10 12 
Equation of time = — 4 10 .9 

Local app. time, < = 4 6 1 .1 
Vol. L— 28 

Altitude of C == /]r= 0*> 30* 20" 

Distance of the nearest limb of the 
sun from the point C= 48® 17 10" 
Semidiameter = 16 1 

2> = 48 88 11 


The sun's declination was 5=4-4° 16' 55", the latitude was 
^ = + 38° 58' 50" ; and hence, by (471), we find 

A = 74° 36' 36" A = 24° 37' 58" 

Befraction and parallax = 1 54 

K = 2A 39 52 

and, by (472), 

a = 43° 35' 6" 

Now, the sun was on the right of the object, and hence 

A'= ^ — a = 31<> 1' 30" 

Therefore, a line drawn on the chart from B on the left of the 
line J5C, making with it the angle 81° 1' 30", will represent the 

280. By two measures of the distance of the ^nfrom a terrestrial 
object. — In the practice of the preceding method with the sextaut, 
it is not always practicable to measure the apparent altitude of 
the terrestrial object. We may then measure the distance of 
the sun from the object at two different times, and, first com- 
puting the altitude and azimuth of the sun at each observation, 
we may from these data compute the altitude of the object and 
the difference between its azimuth and that of the sun at either 
observation, by formulae entirely analogous to those employed 
in computing the latitude and time from two altitudes, Art. 178, 
(304), (305), (306), and (307). 

281. By the azimuth of a star at a given time. — ^When the time is 
known, tlic azimuth of the star is found by (471) : hence we 
have only to direct the telescope of an altitude and azimuth 
instrument to the star at any time, and then compare tlie read- 
ing of its horizontal circle with the computed azimuth. 

This method will be very accurate if a star near the pole in 
employed, since in that case an error in the time will produce a 
comparatively small error in the azimuth. It will be most accu- 
rate if the star is observed at its greatest elongation, as in the 
following article. 

282. By the greatest elongation of a circumpolar star. — At the 
instant of the greatest elongation we have, by Art ] 8, 

cos 8 

sin A = 



in which A is the azimuth reckoned from the elevated pole. At 
this instant the star's azimuth reaches its maximum, and for a 
certain small interval of time appears to be stationary, so that 
the observer has time to set his instrument accurately upon the 

In order to be prepared for the observation, the time of the 
elongation must be (at least approximately) known. The hour 
angle of the star is found by the formula 

tan w 

cos t = — 

tau d 

and from i and the star's right ascension the local time is founds 
Art 65. 

The pole star is preferred, on account of its extremely slow; 

If the latitude is unknown, the direction of the meridian may 
nevertheless be obtained by observing the star at both its eastern 
and its western greatest elongations. The mean of the readings 
of the horizontal circle at the two observations is the reading for 
the meridian. 

283. One of the most refined methods of determining the 
direction of the meridian is that by which the transit instrument 
is adjusted, or by which its deviation from the plane of the 
meridian is measured ; for which see Vol. IE. 

284. At sea, the direction of the meridian, or the variation of 
the compass, is found with sufficient accuracy by the graphic 
process of Art 271. 




285. The term eclipse^ in astronomy, may be applied to any 
obscuration, total or partial, of the light of one celestial body by 
another. But the term sohr eclipse is usually confined to an 
eclipse of the sun by the moon; while an eclipse of the sun by 
one of the inferior planets is called a transit of the planet. An 
eclipse of a star or a planet by the moon is called an ocetdiati<m 
of the star or planet. A lunar eclipse is an eclipse of the moon 
by the earth. 

All these phenomena may be computed upon the same genenl 
principles ; and the investigation of solar eclipses, with which we 
shall set out, will involve nearly every thing required in the 
other cases. 



286. For the purposes of general prediction, and before enter- 
ing upon any precise computation, it is convenient to know the 
limits which determine the possibility of the occurrence of an 
eclipse for any part of the earth. These limits are determined 
in the following problem. 

287. To find xohether near a given conjunction of the sun and moon^ 
an eclipse of the sun loill occur. — In order that an eclipse may occur, 

p. the moon must be near the ecliptic, and, 

therefore, near one of the nodes of her 
orbit. Let NS (Fig. 88) be the ecliptic, .V 
the moon's node, NM the moon's orbit, S 
and M the centres of the sun and moon at 
the time of conjunction in longitude, so 
that MS is a part of a circle of latitude and is perpendicular to 


NS. Let fif' , M', be the centres of the sun and moon when at 
their least true distance, and put 

fi z= the moon's latitude at conjunction = SM^ 

I =z the inclination of the moon's orbit to the ecliptic, 

;i = the quotient of the moon's motion in longitude divided 

by the sun's, 
Iz= the least true distance = /S'JT, 
r = the angle SMS\ 

We may regard NM8 as a plane triangle ; and, drawing M'P 
perpencticular to NSy we find 

and hence 

S'P = |J — 1) tan z' M'P= ^9 — A/9tan r tan Z 

J«= fi» [(A — 1)« tanV + (1 — A tan Ztan r)'] 

To find the value of y for which this expression becomes a mini- 
mum, we put its derivative taken relatively to y equal to zero, 

, X tan I 

tan y= 

(il — 1)* + il» tan' J 
which substituted in the value of 2** reduces it to 

J» = 

iS* (^ - 1)» 

(il — 1)' + A« tan' 7 
If then we assume I' such that 

tanJ'=— ^tanJ (475) 

we have for the least true distance 

I = ficoBr (476) 

The apparent distance of the centres of the sun and moon as 
seen from the surface of the earth may be less than 2* by the 
difference of the horizontal parallaxes of the two bodies : so that 
if we put 

7t = the moon's horizontal parallax, 
Tt' = the sun's " " 

we have 


minimum apparent distance = 2 — (« — «') 

An eclipse will occur when this least apparent distance of the 
centres is less than the sum of the semidiameters of the bodies; 
and therefore, putting 

s = the moon's semidiameter, 
5'= the sun's " " 

we shall have, in case of eclipse, 


2: — (tt — ;r')<S + «' 

fi COSZ' <7r— -7r'4-5 + «' 


This formula gives the required limit with great precision; 
but, since i' is small, its cosine does not vary much for different 
eclipses, and we may in most cases employ its mean value. We 
have, by observation, 

Greatest values. 

Least values. 

Mean valaes. 


5° 20' 6" 

40 57' 22" 

5*^ 8' 44" 


61' 32" 

52' 50" 

57' 11" 






16 46 

14 24 

15 35 


16 18 

15 45 

16 1 





From the mean values of / and X we find the mean value of 
sec /' = 1.00472, and the condition (477) becomes 


^5 < (;r — ;:' + 5 + 5') X 1.00472 

/9<7r--7r'+s + s' + (;r — 7r' + 5+«')X .00472 

where the small fractional term varies between 20" and 30". 
Taking its mean value, we have, with sufficient precision for all 
but very unusual cases, 

,5 < r — r' -1- 5 + s' -;- 25" 



If in this formula we substitute the greatest values of ;r, 5, 
and s'y and the least value of ;r', the limit 

/9 < 1<> 34' 53" 

is the greatest limit of the moon's latitude at the time of cpn- 
j unction, for which an eclipse can occur. 

K in (478) we substitute the least values of ;r, 5, and s\ and 
the greatest value of ;r', the limit 

/9 < I*' 23' 15" 

is the least limit of the moon's latitude at the time of conjunc- 
tion for which an eclipse can fail to occur. 

Hence a solar eclipse is certain if at new moon ^ < 1° 23' 15", 
impossible if j9> 1° 34' 63", and doubtful between these limits. For 
the doubtful cases we must apply (478), or for greater precision 
(477), using the actual values of ;r, ;r', 5, 5', ^, and / for the date. 

Example. — On July 18, 1860, the conjunction of the moon 
and sun in longitude occurs at 2* 19**.2 Greenwich mean time : 
will an eclipse occur ? We find at this time, from the Ephemeris, 

/9 = O*' 33' 18".6 

which, being within the limit 1° 23' 15", renders an eclipse cer- 
tain at this time. 

Having thus found that an eclipse will be visible in some part 
of the earth, we can proceed to the exact computation of the 
phenomenon. The method here adopted is a modified form of 
Bessel's,* which is at once rigorous in theory and simple in 
practice. For the sake of clearness, I shall develop it in a series 
of problems. 

Fundamental Equations of the Theory of Eclipses. 

288. To investigate the condition of the beginning or ending of a solar 
eclipse at a given place on the earth's surface, — The observer sees the 
limbs of the sun and moon in apparent contact when he is situated 
in the surface of a cone which envelops and is in contact with 
the two bodies. "We may have two such cones : 

* See AttronomUehe Naehrkhtenf Nos. 151, 152, and, for the full development, of the 
method with the utmost rigor, Bbssbl's Astronomitche Untenuchungeny Vol. II. 
Hanskn's deyelopment, based upon the same fundamental equations, but theoreti- 
eally less accurate, may also be consulted with advantage : it is given in Attronom, 
Ifaeh., Nos. 889-842. 



First. The cone whose vertex falls between the sun and the 
moon, as at V, Fig. 39, and which is called the penumbral cmte. 
An observer at C, in one of the elements CB V of the cone, see* 
the points A and B of the limbs of the sun and moon in appaient 
exterior contact, which is either the first or the last contact ; that 
is, either the beginning or the ending of the whole eclipse. 

Fig. SB. 

Fig. 40. 

Second. The cone whose vertex is beyond the moon (in the 
direction of the eartli), as at V, Fig, 40, and which is called the 
timbral cone, or cone of total shadow. An observer at C in the 
element CVBA, sees the points A and B of the limbs of the sun 
and mooii in apparent interior contact, wliich is the beginning or 
the ending of annular eclipac in case the obser^-er is farther 
from the moon tlian the vertex of the cone (as in the figure), and 
whicli is either tlie beginning or the ending of tolcd ccli]>se in 
case the obseiror is between the vertex of the cone and the 

If now a plane is passed throngh the point (", at right angles 
to the axis SVD of the cone, its intersection with the cone will 


be a circle (the sun and mooD being regarded ae spherical) whose 
radius, CD, we shall call the radius of the shadow {peniinibral or 
umbra]) for that point. The condition of the occurrence of one 
of the above phases to au observer is, then, that the distance of 
the point of observation from Ike axis of the shadow is equal to the 
radius of the shadow for that point. The problems which follow 
will euable us to traQslate this condition into analytical language. 

289. To find for any given time the position of (he axis of the 
shadow. — The axis of the cone of shadow produced to the celes- 
tial sphere meets it in that point in which the sun would be 
projected upon the sphere by an observer at the centre of the 
moon. Let 0, Fig. 41, be the centre of 
the earth ; S, that of the sun ; M, that of ^'''; *'■ 

the moon. The line MS produced to 
the infinite celestial sphere meets it in 
the common vanishing point of all lines 
parallel to MS; that is, in the point Z, in 
.wliich the line OZ, drawn through the 
centre of the earth parallel to MS, meets 
the sphere. The position of the axis of 
the cone will be determined by tiie right 
ascension and declination of the point Z. 

In order to determine the point Z, let the positions of the sun 
and moon be expressed by rectangular co-ordinates (Art. 32), of 
which the axis of x is the straiglit line drawn through the centre 
of the earth and the equinoctial points, the axis of y the inter- 
flection of the planes of the equator and solstitial colure, and 
the axis of z the axis of the equator. Let x be taken as positive 
towards the vernal equinox ; y as positive towards the point of 
the equator whose right ascension is 90° ; z as positive towards 
the north. 


a, ^, r ^ the right ascension, declination, and distance fVom 

the centre of the earth, respectively, of the moon's 

o',i', r' := the right ascension, declination, and distance fVom 

the centre of the earth, respectively, of the snn's 


The co-ordinates x, y, z will be, by (41), 


Of the iun. Of the mofm. 

r' COS d' COS a' r cos d cos a 

r' COS d' sin a' r cos d sin a 

r' sin d' r sin ^ 

Uow let another system of co-ordinates be taken parallel to th« 
first, the centre of the moon being the origin. The position of 
the sun in this system will be determined by the right ascension 
and declination of the sun as seen from the moon ; that is, by 
the right ascension and declination of the point Z. 
If we put 

tty d = tho right ascension and declination of the point Z, 
G = the distance of the centi*08 of the sun and moon^ 

the co-ordinates of the sun in the new system are 

G cos d cos a 
G cos d sin a 
G sin d 

But these co-ordinates are evidently equal respectively to the 
difference of the corresponding co-ordinates of the sun and moon 
in the first system ; so that we have 

G cos d cos a=.i' cos b' coso' — r cos b cosa 
G cos (i sin a = r' cos b* sin o' — r cos b sin a 
G ^\Xi d = r' sin 5' — r sin b 

which fully determine a, rf, and G in terms of quantities which 
may be derived from the Ephemeris for a given time. 

But, as a and d differ but little from a' and i', it is exp)edient 
to put these equations under the following form. (See the 
similar transformation, Art. 92.) 

G cos d sin (a — <*') = — ^ ^^^ ^ ^^^ (** — **') 

G cos d cos (a — o') = r' cos ^' — r cos b cos (a — o') 

G ^\w d = r' sin ^' — r sin b 

If these are divided by r', and we put 

they become 

g cos d sin {a — a') = — h cos b sin (a — o') 

g cos d cos (a — a') = cos b* — h cos b cos (o — a') 

g ^vci d = sin <5' — b sin ^ 


where the second members, besides the right ascensions and 

declinations, involve only the quantity 6, which may be expressed 

in terms of the parallaxes as follows : 


TT =r the moon's equatorial horizontal parallax^ 

7r'= the sun's " " " 

then we have (Art 89) 

- r sin w' 

f sin TT 
K, further, 

9ro= the sun's mean horizontal parallax, 

and r' is expressed in terms of the sun's mean distance from the 
earth, we have, as in (146), 

and hence 

. , sin TT. 
sm 1^ = — r-^ 

h = ^^^^ (480) 


which is the most convenient form for computing 6, because r' 
and ;: are given in the Ephemeris, and n'o is a constant 

290. The equations (479) are rigorously exact, but as 6 is only 
about jj^, and a — a' at the time of an eclipse cannot exceed 
1° 43', a — a' is a small arc never exceeding 17'', which may be 
found by a brief approximative process with great precision. 
The quotient of the first equation divided by the second gives 

. ,, b cos d BCC d' sin (a — a') 

tan (a — o') = — ^ ^ 

1 — b cos d sec d' cos (tt — o') 

where the denominator diflfers from unity by the small quantity 
b cos 8 sec d' cos (a — a') ; and, since 8 and i' are nearly equal, 
this small difterence may bo put equal to 6, and we may then 
write the formula thus :* 

a — a' = cos d sec d' (a — o') 

1-6 ^ ^ 

♦ DeTeloping the formula for tan (a — o') in series, we have 

, ftcosrfsecfT'sin (a — o') 6' cos' (f sec' d ' sin 2 (a — a') . 

sin 1" 2 sin 1" 

whero the second term cannot exceed 0".04, and the third term is altogether inap- 


If we take cos {a — a') = 1 and cos (a — a') = 1, we have, 
from the second and third of (479), 

g cos d = cos d' — h cos d 

jT sin d = sin d* — b sin d 

g sin ((Z — ^') = — 6 sin {d — d') 
g cos (d — ^') = 1 — h cos (d — d') 

from which follows 

^ l_^cos(^ — d') 

or, nearly,* 

1 — 6^ ^ 

From the above we also have, with sufficient precision for the 
subsequent application of ^, the formula 

The formulae which determine the point Z, together with the 
quantity Gy will, therefore, be 

a = o' cos d sec d' (o — a') 

d = 6' ^(c^-O ^ ^""'^ 

1 — 6 ^ ^ 

and in many cases it will suffice to take the extremely simple 

a = a' — 6 (a — a') d = d'^b^d — d') 

291. To find the distance of a given place of observation from the 
axis of the shadow at a given time, — Let the positions of the sun, 

preciable. The formula adopted in the text is the same as 

a — a'=— 6co8dsec<5'(a — o') (1 — 6)"* 

= — b cos 6 sec <5'(o — o') — 6* cos rf sec 6* (a — o') — &o. 

which, since cos 6 sec i'mny in the second term be put eqnal to unity, dilTers from 
the complete series only by terms of the third order. The error of the approximate 
formula is, therefore, something less than (K'.Ol. 
* The error of this formula, as can be easily shown, will nerer exceed 0^.088. 



Ftg. 41 <«»). 

the moon, and the observer he referred hy rectangular co-ordi- 
nates to three planes passiug through the centre of tbe earth, of 
which the plane of xy shall always be at riglit angles to the axis 
of the shadow, and will here be called the principal plane of refer- 
awe. Let the plane of yz be the plane of the declination circle 
passing through the point Z. The plane 
of xz will, of course, be at right angles 
to the other two. 

The axis of z will then be the line OZ, 
Fig. 41, drawn through the centre of the 
earth parallel to the axis of the shadow, 
and will be reckoned as positive towards 
Z. The axis of y will be the intersection, 
O Y, of the plane of the declination circle 
through Z with the principal plane, and 
will be taken as positive towards the 
north. The axis of x will he the intersection, OJT, of the plane 
of the equator with the principal plane, and will he taken as 
positive towards that point, X, whose right ascension is 90° + a. 

Let M' and S' he the true places of the moon and sun upon 
the celestial sphere, P the north pole ; then, if wc put 

x,y,z ^ the oo-ordinates of the moon, 

we have, by (Art, 31), 

X =^r COB M'X 
y =r COB M' Y 
z ^ r COB M'Z 

which, by the formulte of Spherical Trigonometry appUed to the 
triangles M'PJC, M'PY, M'PZ, become 

= r COB 3 sin {a — a) 

= T [sin 9 cos d — cos i si 

= r [sin i Bin d -|- cos S coa d cos (i 

B(«-a)] I (48! 

8 (a - ay\ ) 

x = rcoBi sin (a — a) \ 

y = r [sin (* — d) cos* i{o — a) -j-sin (3+d)8in'}(o — a)] V (482" 

z = r [cos(a — d)cos'l(a — a) — co8Ca-i-d)sin'i(o — «)] j 

and if the equatorial radius of the earth Is taken as the unit of 


r, X, y, Zj we shall have the value of r, required in these equa- 
tions, by the formula 

r = - — 


The co-ordinates x and y of the sun in this system are the 
same as those of the moon, and the third co-ordinate is z + G; 
but the method of investigation which we are here following 
does not require their use. 

Now let 

f , ly, C = the co-ordinates of the place of observation, 
^ = the latitude of the place, 
^z= the reduced latitude (Art. 81), 
p = the radius of the terrestrial spheroid for the lati- 
tude f, 
fi = the given sidereal time ; 

then, if in Fig. 41 we had taken M for the place of observation, 
M' would have been the geocentric zenith with the right ascen- 
sion fi and declination tp'^ and, the distance of the place from the 
origin being />, we should have found 

i z= p cos f>' sin (pt — a) '\ 

7j = P [sin f ' cos d — cos ^' sin d cos (pt — a)] > (483) 

C = p [sin ^' sin d -f cos ^ ' cos d cos (pt — a)] j 

These equations, if we determine A and B by the conditions 

A sin B = p sin ^ 

A cos B = p cos ^' cos (pt — a) 

may be computed under the form 

S = p cos f ' sin (pi — a) "^ 

ji = Asin(B — d) >(483*) 

C = Aco8lB—d) ) 

The equations (482) might be similarly treated; but the most 
accurate form for their computation is (482*). 

The quantity jte — a is the hour angle of the point Z for the 
meridian of the given place. To facilitate its computation, it is 
convenient to find first its value for the Greenwich meridian. 
Thus, if we put for any given Greenwich mean time T 

fi^ = the hour angle of the point Z at the Greenwich meridian, 
w = the longitude of the given place. 



Fig. 42. 

we have 

PL— a =A*i"- ^ 

To find ix^ we have only to convert the Greenwich mean time T 
into sidereal time and to suhtract a. 

By means of the formnlse (482) and (483) the co-ordinates of 
the moon and of the place of observation can be accurately com* 
puted for any given time. Now, the co-ordinates x and y of the 
moon are also those of every point of the axis of the shadow : so 
that if we put 

A = the distance of the place of observation from the a^s 
of the shadow, 

we have, evidently, 

j«=(x-e)'+(y~^)' (484) 

[The co-ordinates z and {^ have also been found, as they will be 
required hereafter.] 

292. The distance d may be determined under another form, 
which we shall hereafter find useful. Let M'j 
Fig. 42, be the apparent position of the moon's 
centre in the celestial sphere as seen from the 
place of observation ; P the north pole ; Z the 
point where the axis of the cone of shadow 
meets the sphere, as in Fig. 41 ; Jfj, CJ, the 
projections of the moon's centre and of the 
place of observation on the principal plane. 
The distance C^M^ is equal to J, and is the 
projection of the line joining the place of 
observation and the moon's centre. The plane by which this 
line is projected contains the axis of the cone of shadow, and 
its intersection with the celestial sphere is, therefore, a great 
circle which passes through Z, and of which ZM' is a portion. 
Hence it follows that C^M^ makes the same angle with the axis 
of 2/ that M^Z makes with PZ: so that if we draw C^N and M^N 
parallel to the axes of y and z respectively, and put 

Q = PZM* = NC,M^ 

we have, from the right triangle C^M^N^ 

J sin e = a: — f 1 

jcose = y-^ I (485) 

the sum of the squares of which gives again the formula (484). 


293. To find the radius of the shadow on the principal plane y or on 
any given plane parallel to the principal plane, — This radius is evi- 
dently equal to the distance of the vertex of the cone of shadow 
from the given plane, multiplied by the tangent of the angle of 
the cone. In Figs. 39 and 40, p. 440, let EF be the radius of 
the shadow on the principal plane, CD the radius on a parallel 
plane draw^i through C Let 

H = the apparent semidiameter of the sun at its mean dis- 

k = the ratio of the moon's radius to the earth's equatorial 

/ = the angle of the cone = EVF, 

c = the distance of the vertex of the cone above the princi- 
pal plane = FF, 
C = the distance of the given parallel plane above the prin- 
cipal plane = DF, 
I = the radius of the shadow on the principal plane = EF, 

L = the radius of the shadow on the parallel plane == CD. 

If the mean distance of the sun firom the earth is taken as 

unity, we have 

the earth's radius = sin r^ 

the moon's radius == k sin r, = MB, 

the sun's radius = sin J3 = SA, 

and, remembering that G = r'g found by (481) is the distance 
MSy we easily deduce from the figures 

,i^j.^«-ng±Asin., ^^gg^ 


in which the upper sign corresponds to the penumbral and the 
lower to the umbral cone. 

The numerator of this expression involves only constant quan- 
tities. According to Bessel, i2'=959".788; Excke found 
;r^ = 8".57116 ; and the value of i, found by Burckhardt from 
eclipses and occultations, is k = 0.27227 ;* whence we have 

log [sin JE + k sin rj = 7.6688033 for exterior contacts, 
log [sin H — /: sin ;rj = 7.6666913 for interior contacts. 

* The ralue of k here adopted is precisely that which the more recent inrestiga- 
tion of OuDEMANB {Attroti. Naeh., Vol. LI. p. 80) gives for eclipses of the sun. 
For occultations, a slightly increased value seems to be required. 


ISoWy taking the earth's equatorial radius as unity, we have 


MF = z (Art. 291) 
and hence 

c = z±— (487) 

sin/ ^ ^ 

the upper sign heing used for the penumbra and the lower for 
the umbra. 
We have, then, 

1 = c tan/ = z tan/ ± k sec/ 
L= {c — C) tan/ = I — C tan/ 

I (488) 

For the penumbral cone, c — f is always positive, and there- 
fore L is positive also. 

For the umbral cone, c — f is negative when the vertex of 
the cone falls below the plane of the observer, and in this case 
we have total eclipse : therefore for the case of total eclipse we 
shall have L = {c — f) tan /a negative quantity. It is usual to 
regard the radius of the shadow as a positive quantity, and 
therefore to change its sign for this case ; but the analytical dis- 
cussion of our equations will be more general if we preserve 
the negative sign of i as the characteristic of total eclipse. 

When the vertex of the umbral cone falls above the plane of 
the observer, L is positive, and we have the case of annular 

For brevity we shall put 

t = tan/ "I 

l = ic V (489) 

L=l^i: ) 

294. The analj/iical expression of the condition of beginning or 
ending of eclipse is 

or, by (484) and (489), 

(^ - O' + (y - >?)• =(i- i:y (490) 

It is convenient, however, to substitute the tvvo equations 
(485) for this single one, after putting L for J, so that 

(I — iC) sin Q = X — c 
(I — iZ) cos Qz=zy — Ti 
Vol. L— 29 

} (491) 


may be taken as the conditions which determine the beginning 
or ending of an eclipse at a given place. 

The equation (490), which is only expressed in a different form 
by (491), is to be regarded as the fundamental equation of the 
theory of eclipses. 

295. By Art. 292, so long as J is regarded as a positive quan- 
tity, Q is the position angle of the moon's centre at the point Z; 
and since the arc joining the point Z and the centre of the moon 
also passes through the centre of the sun, Q is the common 
position angle of both bodies. 

Again, since in the case of a contact of the limbs the arc 
joining the centres passes through the point of contact, Q 
will also be the position angle of this point when all three 
points — ^sun's centre, moon's centre, and point of contact — ^lie 
on the same side of Z. In the case of total eclipse, however, 
the point of contact and the moon's centre evidently lie on 
opposite sides of the point Z; and if I — i^ in (490) were a 
positive quantity, the angle Q which would satisfy these equa- 
tions would still be the position angle of the moon's centre, but 
would differ 180° from the position angle of the point of con- 
tact. But, since we shall preserve the negative sign of / — if 
for total eclipse (Art. 293), (and thereby give Q values which 
differ 180° from those which follow from a positive value), the 
angle Q will in all cases be the position angle of the point of coniacL 

296. The quantities a, rf, x, y, /, and i may be computed by 
the formula (480), (481), (482), (486), (487), (488), for any given 
time at the first meridian, since they are all independent of the 
place of observation. In order to facilitate the application of 
the equations (490) and (491), it is therefore expedient to com- 
pute these general quantities for several equidistant instauts 
preceding and following the time of conjunction of the sun and 
moon, and to arrange them in tables from which their values 
for any time may be readily found by interpolation. 

The quantities x and y do not vary uniformly ; and in order to 
obtain their values with accuracy from the tables for any time, 
we should employ the second and even the third differences in 
the interpolation. This is effected in the most simple manner 
by the following process. Let the times for which x and y have 
been computed be denoted by T^ — 2*, T^ — 1*, T;, T^ + 1*, 



T^ + 2*, the interval being one hour of mean time ; and let the 
values of x and r/ for these times be denoted by a:_2, x^i, &c., 
y_2, y-i, &c. Let the mean hourly changes of x and y from the 
epoch T^ to any time T=Tq + rhe denoted by x' and y'. Then 
the values of x' and y' for the instants T^ — 2*, T^ — 1*, &c. will 
be formed as in the following scheme, where c denotes the third 
difference of the values of x as found from the series x_2> ^—h ^^^ 
according to the form in Art 69, and the difference for the 
instant T^ is found by the first formula of (77). The form for 
computing y^ is the same. 






iix,— x_,) 



Xf, — X_i 



l(x, — z_,) — 




x^ — x^ 

r.+ 2» 


I (x, — x,) 

If then we require x and y for a time ST = TJ^ + r, we take 
xf and y' from the table for this time, and we have 

a: = Xfl 4" ^^ 

y = yo + y'^ 

297. Example. — Compute the elements of the solar eclipse of 
July 18, 1860. 

The mean Greenwich time of conjunction of the sun and 
moon in right ascension is July 18, 2* 8"* 56*. The computation 
of the elements will therefore be made for the Greenwich hours 
0, 1, 2, 8, 4, and 5. For these hours we take the following 
quantities from the American Ephemeris : 

M For the Moon. 

Greenwich meui 




July 18, 0» 

116^ 44' 24''.30 

21° 52' 20".3 

59* 45".80 


117 21 59 .10 

42 82 .8 

47 .13 


117 69 30 .46 

82 86 .4 

48 .44 


118 86 58 .85 

22 31 .2 

49 .72 


119 14 22 .65 

12 17 .2 

50 .98 


119 51 48 .85 

1 54 .6 

62 .22 


^'iu.«^ is"^ 


fT Il#^* 







iu-j > 


^ ^ ^^^^1 

i!K -I'.f^ 

JLi*= ~ w-' JO 

J- JUvBHlt ■» 


* 1: ^> 

■ji ll!^ -za. 



-t «i :^ 

rr i « 




' 14 r- 

in ij TTj 




> « -» 

>i ? .^ 



li :^ .Ki 

54 41 J4 


, -^ ',* i*~" T. dfr- •' '•*■ 

T, :: Ltl^^I 'V\^ Oft iLcT^ 

arai. ?:c 

f'*i& r 


1 // 

i:»r an Tj = 5.»51*94 

— »* cr. aearlTT. i = »' — 5 » — 

i = ?' — 6i-» — r, 

// I // 



r n't ft (d d) t'Jm^ i(a 
r cJfH (d d) iihH^ i(a 

a) -;- r sin (^ — J) sin' } (o — a) 
aj — r cos (^ — d) sin' \{a — a) 

III. TJm? an^I<j of the cone of shadow and the radius^f the 

for pmuml/ra: or exterior rontacU. 


hiu f : 

For umbra: or mierior contaeti 


. __ [7.666691] 

C'^- z -\ , log k = 9.435000, c = z 

Hin / sin/ 

I = tun/ 
i= ic 

i == tan f 
l = %c 



IV. The values of a, rf, x, y, log i, and I, will then be tabulated 
and the differences x' and y' formed according to Art. 296. 

I give the computation for the three hours 1*, 2*, and 3*, 
in extenso. 

I. Elements of ike point Z. 


a — a 
<J— (J' 

log ooseo TT = log r 

ar. CO. log r' 

Constant log sin tt^ 


ar. CO. log (1 — b 

log cos 

log sec (V 

log (a — o') 

log (a — tt') 

a — a' 


(1) + (2) 

log (.5 




log (1 — 6) = log g 



0® 40^ 13". 40 

-^ 6'12".69 

-f 46 3 .38 

-f- 85 33 .80 














nd. 38263 




4- 5". 66 

4- 0".78 







6". 88 

— 6".08 

118° 2' 18". 16 

118° 4'48".87 

20 67 28 .04 

20 66 67 .67 




+0° 29' 44".58 
+ 25 65 .45 



— 4".19 



nO. 56428 

— 8".67 

118° r 9".68 
20 66 82 .08 


n. Co-ordinates x, y, and z. 

a — a 

log sin (tt — a\ 

log cos o 

log r COS 6 sin (a — a) = log x 


logcos'^ftt — a) 
log 8in(d — d 
log(8)=logr8in(<J— </)co8«}ra— a 

log sin' J (a — a 
log sin (6 -\- d) 
log (4) =log r sin (<J -f d) sin* }(*—«) 

(8) + (4) 


log cos [6 — d\ 
log (5)= log r cos (d — rf)co8'J(o — a) 

log cos (c5 -f ^) 
log (6) =logr cos (6 -f- rf)8in* \ (a — a) 
log [(6) - (6)] = log z 

— 0° 40* 19".06 
+ 45 9 .76 
42 39 66 .84 
n9. 796961 7 


-f 0.755402 




— 0® 5'18".42 

4- 85 38 .88 

42 29 33 .97 



n8. 9097909 



-j-0. 696058 




4-0° 29* 48". 77 
4- 25 59 .12 
42 19 8 .28 








HI. Log i and Z, for exterior contacts. [Constant log = 7.668803] 



log r'sf — 

log Bin/ 
log Bee/ 


k coseo / 

log [2 


* coBec/] 

= logc 

log tai 


= log t 

log ie 




























Log i and I for interior contacts. [Constant log = 7.666691] 

Const. — log r*^ = log sin/ 

log sec/ 
log k cosec / 
log [« — k cosec/] = log e 
log tan/ = log t 

log te == log / 




n7. 964788 












IV. The computation being made for the other hours in the 
same manner, the results are collected in the following tables. 







Exterior Oontacta. 

Interior CooUcta. 





lOf » 


118 2 18 .16 

4 43 .87 

7 9.58 

9 85 .27 

12 .95 

20° 57' 48".60 
67 23 .04 
56 57 .57 
56 82 .08 
56 6 .68 
66 41 .06 



— 0.008960 

64 , 

57 ; 











— 27 

— 87 







— 0.081244 
-1- 0.464044 
4- 1.009245 
-h 1.664284 

4- 0.646297 

-f- 0.917040 
l-f- 0.756742 
l-f 0.596076 
-j- 0.278704 
!-f 0.1 12089 

— 0.160298 

— 0.161019 

— 0.161852 

— 0.161666 






For the values of the hourly differences of x and y, we find 
from the above, by Art. 296, 




log JC* 






— 0.160483 





— 0.160607 


r, — 2 








— 0.161019 





— 0.161186 







and for any given time T = TJ^ + r, we have 

x = — 0.081244 + a/r 
y = + 0.596075 + }fx 

} (492) 

Finally, to facilitate the computation of the hour angle 
fi — a=^ fx^ — (a (Art, 291), we prepare the values of /sij for each 
of the Greenwich hours. Thus, for T = 1*, we have 

From the Ephemeris, July 18, 1860, 

Sid. time at mean noon = 7* 46* 4* .03 

Sid. equivalent of P mean t. = 1 9 .86 

Greenwich sid. time = 8 46 13 .89 

" <* " in arc, = 131* 33' 28".35 

a =118 2 18 .16 

Ml = 13 31 10 .19 

Thus we form the following table, to which is also added for 
future use the value of the logarithm of 

fi' = the hourly difference of ii^ in parts of the radius; 

log Ai' = log 54002".15 sin 1" 
= 9.417986 


Hourly diff. 


358" 31' 8".0 


13 31 10 .2 


28 31 12 .3 



43 81 14 .4 


58 81 16 .6 


78 81 18 .7 

I proceed to consider the principal problems relating to the 
general prediction of eclipses, in which the preceding results 
will be applied. 


Outline of the Shadow on the Surface of the Earth. 

298. To find the outline of the moon's shadow upon the earth at a 
given time, — This outline is the intersection of the cone of shadow 
with the earth's surface ; or, it is the curve on the surface of the 
earth from every point of which a contact of the son's and 
moon's limbs may be observed at the given time. Let 

T = the given time reckoned at the first meridian, 

and let a, rf, x, y, ?, and log i be taken firom the general tables 
of the eclipse for this time. Then the co-ordinates f , 37, f of any 
place at which a contact may be observed at the given time must 
satisfy the conditions (491), 

} (493) 

(I — tC) sin e = a: — ? 

{l — tC) cos § = y — 17 

^ = the hour angle of the point Z^ 

w=z the west longitude of the place; 
then we have 

^ z=z fi — a =: /l^ — m 

and the equations (483) become 

S = p cos ^'sin 6 \ 

7j =z p sin /cos d — /» cos f' sin d cob ^ > (494) 

C = /t> sin f ' smd-^ p cos / cos d cos * J 

The five equations in (493) and (494) involve the six variables 
f, 7;, ^, f ', «?, and §, any one of which may be assumed arbi- 
trarily (excluding, of course, assumed values that give impossible 
or imaginary results) ; then for each assumed value of the arbi- 
trary quantity we shall have five equations, which fully deter- 
mine five unknown quantities, and thereby one point of the re- 
quired curve. I shall take Q as the arbitrarj^ variable. 

In the present form of the equations (494), they involve the 
unknown quantity />, which being dependent upon ip' oannot be 
determined until the latter is found. This seems to involve the 
necessity of at first neglecting the compression of the earth, by 
putting p = Ij and after an approximate value of ^' has been 
found, and thereby also the value of />, repeating the computation. 
But, by a simple transformation given by Bessel, this double 
computation is rendered unnecessary, and the compression of 


the earth is taken into account from the beginning. If ip is the 
geographical latitude, we have (Art 82) 

, cos ip . , sin cp (1 — ee) 

^ cos f>= —, V- ^sm/= ^ 

^'(l — ee sin' f>) ^'(l — ee sin' <p) 

in which 

log ee = 7.824409 log i/(l — ec) = 9.9985458 

If we take a new variable ^ j, such that 

cos ip 

cos «Pi = —-' —, 

^* |/(1 — ee sin* ^) 

we shall have 

//-I 1 N sin^T/(l— ee) 

|/(1 — ee sm' f>) 

cos f^^p COS ^' 

p/(l — ee) sin ^^^ p sin f>' 

tany= ^^^> 

1/(1 -ee) 

Hence the equations (494) become 

f = cos f , sin * 

ly ^ sin f J cos (Z |/(1 — ee) — cos ^^ sin d cos ^ 

C = sin ^j sin (Z p/(l — ee) + cos ^^ cos d cos »> 


p^ sin Jj =: sin d p^ sin ^, = sin d |/(1 — ^) ) 

/9j cos dj = cos d |/(1 — ee) />, cos d^ = cosd J ^ ^ 

The quantities />i, dj, />„ d,* ^^^7 ^^ computed for the same times 
as the other quantities in the tables of the eclipse, and hence 
obtained by interpolation for the given time. The factors 
/Oj and />, will be sensibly constant for the whole eclipse. We 
now have 

f = cos f J sin * 

ly = /9j sin f>j cos d^ — /9, cos ^ i sin d^ cos t> 
C = /o, sin ^, sin (f, + />, cos ^^ cos d^ cos i> 
Let us put 


and assume (^„ so that 

? + '?,• + C,« = 1 (496) 


or, which is equivalent, let us take the system 

f = cos f , sin * 

i^j = sin f J cos dj — cos f>j sin rf, cos ^ ^ (497) 

C, = sin f , sin d^ + cos ^ ^ cos d^ cos * 

The quantity f i differs so little from f that we may in practice 
substitute one for the other in the small term i^ ; but if theo- 
retical accuracy is desired we can readily find f when fj is 
known ; for the second and third of (497) give 


^j COS * = — i^j sin d^ + C, cos d^ 
sin f^ = 1}^ cos dj + Ci sin d^ 

which substituted in the value of (^ give 

C = />, C, cos (d, - d,) - p, 71^ sin (rf, - d,) (498) 

Our problem now takes the following form. We have first 
the three equations 

(i-i:,)cose = y-^,,, V (499) 

f* + 'yi' + ^1* = 1 ) 

which for each assumed value of Q determine f , tj^j and f j. Then 

we have 

cos f>j sin * = f 'J 

COB ^j cos d = — i^j sin rfj + Ci cos d^ V (600) 

sin fj= i^j cos rfj + Ci sin d^ ) 

which determine <p^ and A Then the latitude and longitude of 
a point of the required outline are found by the equations 

tan sp = ^^"^' 0, = « — * (501) 

|/(1 -— ee) 

To solve (499), let ^ and y be found by the equations 

sin /9 sin T' = X — / sin Q = a 

then we have 

sm fi cos r = — = ^ 

$ =: sin iS sin ^^ + iCi sin Q 
ly^ = sin fi cos 7' + iC^ cos Q 


where we have omitted /o^ as a divisor of the small term i!^^ cos Q, 
since we have very nearly p^ = 1. Substituting these values in 
the last equation of (499), we find 

Cj* = C08»/9 — 2i> Bin /9 cos (C — r) — (»Ci)' 

Neglecting the terms involving i* as practically insensible, this 


Ci= ± [cos fi — I sin fi cos (Q — r)] 

In order to remove the ambiguity of the double sign, let us put 

Z = the zenith distance of tJie point Z(Art. 289) ; 
then, since & = fi — a is the hour angle of this point, we have 

cos Z == sin f sin d -f cos f cos d cos ^ 
which by means of the preceding equations is reduced to 

C08 2 = c. /., ^ (503) 

sm ^j 

Hence cos Z and (^^ have the same sign. 

But, in order that the eclipse may be visibk from a point on 
the earth's surface, we must, in general, have Z less than 90° ; 
that is, cos Z must be positive, and therefore f ^ must be taken 
only with the positive sign. The negative sign would give a 
second point on the surface of the earth from which, if the earth 
were not opaque, the same phase of the eclipse would also be 
observed at the given time. In fact, every element of the cone 
of shadow which intersects the earth's surface at all, intersects 
it in two points, and our solution gives both points. 

K we put 

i cos (Q^r) 

sin 1" 
we have 

Cj = cos fi — sin /9 sin e 


or, with sufficient accuracy, 

C, = cos (fi + e) (505) 

Thus, fi and y being determined by (502), f j is determined by 
(504) and (505) : hence also f and ij^ by the equations 


The problem is, therefore, fully resolved ; but, for the conve- 
nience of logarithmic computation, let c and C be determined 
by the equations 

'^"°^='' 1(507) 

c cos (7 = Ci J 

then the equations (500) become 

cos sp, sin 1^ = ^ 1 

cos f J cos d = (? cos (C + (Zj) > (508) 

sin ^j= c sin ((7 + d^) ) 

The curve thus determined will be the intersection of the 
penumbral cone, or that of the umbral cone, with the earth's 
surface, according as we employ the value of I for the one or the 

299. The above solution is direct, though theoretically but 
approximate, since we have neglected terms of the order of t*. 
It can, however, readily be made quite exact as follows. We 
have, by substituting the values of ^^ and tJj^ in (498), and neg- 
lecting the term involving the product isin({^ — d^), which is 
of the same order as i*, 

C = /o, cos (/? + «) — /», sin /9 cos j^ sin (d^ — d^) 

and, putting 

e' = (tfj — d^) cos 7' 

we have, within terms of the order i', 

C = />, cos (/5 + c + e') (509) 

The substitution of this value of ^ in the term if involves only 
an error of the order t^, which is altogether insensible. The 
exact solution of the problem is, therefore, as follows. Find ^ 
and Y for each assumed value of Q, by the equations 

sin fi sin ]r =^ oc — I sinQ = a 

. ^ y ^ cosO , 

sm 13 cos r = — —= 

Pi Pi 

then e and e' by the equations 

sm 1" 


Find ^' and y' by the equations 

sin /?' sin / = a + i>, cos (/9 -f- e + c') sin Q = $ 

. ^- , 1 . t>, cos (^ + c + c') cos Q 
sin /9' cos •/= b + -^^-? ^^—5- — -!- — ^ — = 19, 


then we have, rigorously, 

Cj = cos /9' 

and these values of f , i^j, and (^^ may then be substituted in (500), 
which can be adapted for logarithmic computation as before.* 

300. It remains to be determined whether the eclipse is begin- 
ning or ending at the places thus found. A point on the earth's 
surface which at a given time T is upon the surface of the cone 
of shadow will at the next consecutive instant T -{- dT be 
within or xcithout the cone according as the eclipse is beginning or 
ending at the time T; the former or the latter, according as the 
distance A = i/[(a: — f)* + {y — yjf] becomes at the time T + dT 
less or greater than the radius of the shadow I — i^. In the case 
of total eclipse I — i^ is a negative quantity, but by comparing 
J* with (I — i(^y we shall obtain the required criterion for all 
cases ; and, therefore, the criterion of beginning or ending, either 
of partial or of total eclipse, will be the negative or positive value 
of the difterential coeflSicient, relatively to the time, of the 

or the negative or positive value of the quantity 

'^ ^\dT dTr^ ^'\dT dT} ^ ^\dT dTl 

* In this problem, as weU as in most of the subsequent ones, I haye not followed 
Besskl's methods of solution, which, being mathematically rigorous, though as 
simple as such methods can possibly be, arc too laborious for the practical purposes 
of mere prediction. As a refined and exhaustive disquisition upon the whole theory, 
Bessel's Analyse der Finstemuie, in his Aitronovmche Untertuchungen, stands alone. 
On the other hand, the approximate solutions heretofore in common use are mostly 
quite imperfect ; the compression of the earth, as well as the augmentation of the 
moon's semidiameter, being neglected, or only taken into account by repeating the 
vhole computation, which renders them as laborious as a rigorous and direct method. 
I haye endeaTored to remedy this, by so arranging the successive approximations, 
when these are necessary, that only a small part of the whole computation is to be 
repeated, and by taking the compression of the earth into account, in all cases, from 
the commencement of the computation. In this manner, even the first approxima- 
tions by my method are rendered more accurate than the common methods. 


where we omit the insensible variation of i. For brevity, let as 

dx dv 
write x\ y\ &c. for -77=7, -r—-, &c. and denote the above quantity 

by P; then, after substituting the values of x — f = (Z — i f ) sin §, 
jf — ij = {I — i^) cos Qj we have 

P= L [(x'- e')8^n « + (y^- V)co8C - (/'- tC')] 
in which L=l — i(^. K we put 

P' = (a/ — e') sin e + (y — ,') cos c — (r - tr) (6io) 

we shall have 

The quantity P will be positive or negative according as L and 
P' have like signs or different signs. 

For exterior contacts, and for interior contacts in annular 
eclipse, L is positive (Art. 293), and hence for these cases the eclipse 
is beginning or ending according as P' is negative or positive; but for 
total eclipse, L being negative, we have beginning or ending 
according as P' is positive or negative. 

We must now develop the quantity P'. Taking one hour as 
the unit of time, x', y', 1% f ', 7j\ ^', will denote the hourly changes 
of the several quantities. The first three of these may be 
derived from the general tables of the eclipse for the given time; 
but f ', r/, ^' are obtained by differentiating the equations (494), 
in which the latitude and longitude of the point on the earth's 
surface are to be taken as constant. Since 1? = ft — a>, we shall 

have —p=f = -j^ ; and hence, putting 

we find 

y=,*^8inr d'= — sinr 

dT dT 

r' = IX p cos f>'co8 ^ = pJ ( — yj Bin d + C cos d) 
= fi' [ — y sin d + C cos d -]- (I — iC) sin d cos Q] 

y = ju' $ sin d — d'Z 

= /A Ix sin d — (I — I'C) sin d sin Q] — d'C 

C'=— /t'eco8rf + d'li 
= p! [—X coBd + (I — tC) cos 4 sin Q] + d' [t/ -- (I — tC) cos Q} 


Substituting these values in (510), and neglecting terms involving 
I* and id' as insensible, we have 

P'=z a' — y cos Q + </ smQ — C (m' cos ^ sin Q — d' cos Q) 

in which a', 6', and c', denote the following quantities: 

a'= — r — ju' ix cos d "j 

h'= — y'+ii'x sin d V (511) 

(/ = xf -{- //y sin d -\~ fi' il cos d ) 

The values of these quantities may be computed for the same 
times as the other quantities in the eclipse tables, and their 
values for any given time will then be readily found by interpo- 
lation. For any assumed value of Qj therefore, and with the 
value of Z found by (509), the value of P' may be computed, and 
its sign will determine whether the eclipse is beginning or 
ending. In most cases, a mere inspection of the tabulated values 
of a', 6', and c', combined with a consideration of the value of 
Qy will suffice to determine the sign of P' ; but when the place 
is near the northern or southern limits of the shadow, an accu- 
rate computation of P' will be necessary; and, since other appli- 
cations of this quantity will be made hereafter, it will be proper 
to give it a more convenient form for logarithmic computation. 

csinJS7 = y fBmF=d' lr512^ 

eQO%E=cf /cos^=ju'cos d j^ ^ 

then we have 

P' = a' + « sin (C — -^ — C/ sin (C — F) (513) 

Since a' and -Pare both very small quantities, and a veiy precise 
computation of P' will seldom be necessary when its algebraic 
sign is alone required, it will be sufficient in most cases to neglect 
these quantities, and also to put f j for f , and then we shall have 
the following simple criterion for the case of partial or annular 
eclipse : 

If e sin (Q — ^) < Ci/ sin §, the eclipse is beginning. 
If e sin (Q — i?) > Cj/ sin Q, the eclipse is ending. 

For total eclipse, reverse these conditions. 

801. In order to facilitate the application of the preceding as 
well as the subsequent problems, it is expedient to prepare the 
values of d^^ log p^^ d^, log />„ a', fc', c', e, -E, /, 2^, and to arrange 
them in tables. 

'*7 » 

*■ "XJ 

3b 3«8t wilii^ 

jr* 4- ?•• 

i. 3 ^ 

L. S^ -' 




i** ■*.:•»< r 


lu** <»•.'. -• 


l«!*Sf- i^J. • 


- 1 --11 ttTTTTtt. Aif* fram 

_ I 

~ '.^GZiH 






•• '' " »• , ^■•l' —1,* ""li"**^ ""J" l' 




- ■ i^r^v 


^r n-iTTif riTCSB-fc 

— - 





1 it»«<-ii _ 
1 •»•♦■♦_ 

T:-<r vaij"* of '. K. f. F. i'ffT rjiirrtr r:#*»w.^*v. dt:^iuc>rd from 
TL<r--r v^; j*'*- *>f 6' ajid c'. a;.d from (f'= — 25 ".o sin l'\bv lolik, 








4° 33' 21" 


— 0' 1' 44" 



9 22 25 





14 17 17 





19 13 48 





24 7 46 





28 55 7 




302. To illustrate the preceding formulse, let lis find some 
points of the outline of the penumbra on the earth's surface at 
the time T= 2* 8* 12*. For this time, we have 

a; = — 0.00672 ^ogp^= 9.99873 log i = 7.66287 

y = + 0.57409 d^= 21<» 0' 45" 

I = + 0.53673 ti^= 30 34 13 

Let us find the points for Q = 50° and Q = 800°. The com- 
putation may be arranged as follows : 


By (502) : 

Hence by (504) : 

By (506) : 
By (507) : 


a = sin /9 sin y 
6 = sin ^ cos Y 



By (505) : log C, = log cos (/9 + e) 

iCiSin Q 
iZi cos Q 

17, = log c sin C 

Cj = log c cos C 



log cos f J sin i> 

log cos ^j cos »^ 

log tan d 

log cos f J 

log sin ip^ 

log tan ip^ 

log i/(l — ee) 

log tan f 

Vol. L*30 

By (508) : log ? 
logc cos (C+ rfj) 

logc sin (C+ <^i) = 




h 0.45810 

+ 0.22975 

- 0.30662 

61<» ir 52" 

56<» 12' 16" 

28 28 52 

33 27 7 

5 43 

— 6 59 

28 23 9 

33 20 8 



h 0.00310 

— 0.00333 

- 0.00260 

+ 0.00192 


+ 0.45477 

+ 0.23235 

+ 0.30854 







14^ 47' 39" 

20^16' 9" 

35 48 24 

41 16 54 

















— 29<» 20' 20" 

34^*11' 46" 

59 54 33 

356 22 27 

32 15 3 

36 4 40 



To find whether the eclipse is beginning or ending at these 
places, we have, from the table on p. 465, for T=^ 2* 8"* 12*, 




14^ 58' 

Q E 

85 2 

285<> 2' 

log e sin (C — E) 





log Ci/ sin Q 



At the first point, therefore, we have e sin (Q — -E) > fj/sin §, 
and the eclipse is ending. At the second point, we have 
6 sin(§ — E) < Qy^f&in Q^ and the eclipse is beginning. 

Rising and Setting Limits. 

303. To find the rising and setting limits of the eclipse. — ^By these 
limits we mean the curves upon which are situated all those points 
of the earth's surface where the eclipse begins or ends with the 
sun in the horizon. It will be quite sufficient for all practical 
purposes to determine these limits by the condition that the 
point Z is in the horizon. This gives in (503) cos^=0, or 
Ci = 0, and, consequently, by (496), we have 

as the condition which the co-ordinates of the required points 
must satisfy. 

Now, let it be required to find the place where this equation 
is satisfied at a given time T. Let x and y be taken for this 
time, then we have, by putting Ci= in (499), 

? sin Q = X — ^ 

? cos § = y — ly 

m sin Jtf" = a: 

m cos M =^y 

then, from the equations 

? sin Q = m %\n M — p sin y 
I cos Qz=vi cos M — p cos y 

we deduce, by adding their squares, 

Z* = m' — 2mp cos (3f — f) ^ p* 

2 sin«}(Jtf— y) = 1 — cos (3f — r) = ^— (^ — 1^)' 


I? sin z' = f 
p cos r ^=^ 

] (516) 
} (516) 


If then we put k=^ M — r? we have 

8in i>l = dz ^p + m--;2j^-m+j>)1 


Y -= M :^k 

Hence we have 

m which \X may always be taken less than 90°, but the double 
sign must be used to obtain the two points on the surface of 
the earth which satisfy the conditions at the given time. 

In this formula, m, 3/, and I are accurately known for the 
given time, but p is unknown. It is evident, however, from 
(514) and (515), that we have nearly J» = 1, and this value may 
be used in (517) for a first approximation. To obtain a more 
correct value of 7*, let us put f = sin y'\ then, by (514), we have 
1^1= cos 7^, and, consequently, since tj^ p^jj^, 

p Bin y = sin / 
p cos Y z=z p^ cos f 

tan •/ z= p^ tan y 

sin/ p.cos/ } (518) 

p = — — — = *-^ 2- I 

sin Y cos Y 

and with this value of p the second computation of (517) will 
give a very exact value of 7*. With this second value of 7* a still 
more correct value of p could be found; but the second approxi- 
mation is always sufficient. 

With the second value of 7*, therefore, we find the final value 
of f by the formula 

tan / = /Oj tan y 

and then, substituting the values f = sin y\ t^^ = cos y^, ^j == 0, in 
(500), we have, for finding the latitude and longitude of the 
required points, the formulse 

cos f J sin ^ = sin / 
COB f , cos & = — cos / sin d^ 
sin ^j = cos / cos rf, 


a> = /I, — 1* tan f == ^ — 

1/(1 — ee) 

In the second approximation, we must compute X and y by 
(517) separately for each place. 


304. Tlie sun is rising or setting at the given time at the 
places thus deteniiined, according as ^ (which is the hour angle 
of the point Z) is between 180° and 360° or between 0° and 180°. 

To determine whether the eclipse is beginning or ending, we 
may have recourse to the sign of P' (513); and it will usually be 
sufficient for the present problem to put both a' and f = in 
that expression, and then the eclipse is beginning or ending 
according as sin {Q — E) is negative or positive. Now, by (516), 
we find 

I sin (Q — ^) = m sin (ilf — E) — p sin (r — JS) 

Hence, for points in the rising or setting limits, 

If m sin (ibf — E) <Cp sin (y — E), the ecHpse is beginning, 
If m sin (M — E) ^ p sin (jr — E), the ecHpso is ending. 

305. In order to apply the preceding method of determining 
the rising and setting limits, it is necessary first to find the 
extreme times between which the time T is to be assumed, or 
those limits of T between which the solution is possible. The 
two solutions given by (517) must reduce to a single one when 
the surface of the cone of shadow has but a single point in 
common with the earth's surface, — i.e. in the case of tangeney of 
the cone and the terrestrial spheroid. Now, the two solutions 
reduce to one only when ^ = 0, and both values of y become == 3f; 
but if k = 0, the numerator of the value of sin iX must also be 
zero; and hence the points of contact are determined by the 

I -\- VI — p =z and I — m -]- p = 

or bv the conditions 

VI = p -\- I and m = p — I 

There may be four cases of contact, two of exterior and two of 
interior contact. The two exterior contacts are the first and last, 
or the beyitudvg and the end of the eclipse generally ; the axis of the 
shadow is then without the earth, and therefore we must have 

for these cases m = \^dr + y' = p -{- L 

The first interior contact corresponds to the last point on the 
earth's surface where the eclipse ends at sunrise ; the second, 
to the first point where it begins at sunset. But these interior 


contacts can occur only when the whole of the shadow on the 
principal plane falls within the earth, and for these cases, there- 
fore, we must have m = p — L 

For the beginning and end generally we have, therefore, by 


(p -{- 1) »\n M =z X 

(p -f- I) cos M=y 
Let The the time when these conditions are satisfied, and put 

in which Tq is the epoch of the eclipse tables, for which the 
values of x and y are x^ and y^. Then, x' and ^' being the mean 
hourly changes of x and y for the time T, we have 

} (520) 


m^ sin M^ = x^ n sin N = x' 

m^ cos M^ = yo ^ cos N=\f 

the above conditions become 

(l> -f I) sin M= Mq sin M^ -\-t ,n»\n N 
(p -|- cos M= hIq cos M^-^r ,n cos N 

(p + I) sin (3f — iV) = m^ sin (M^ — iVT) 

iP + cos (Jf — iVT) = m^ cos (M^ — N) +nT 
so that, if we put JK* — iV = -v^, we have 

sin 4, = ^o sin (M, - N) 

p + l 

r=P±Icos^-'^cos(M,-N) ) (^21) 

T= T^ + T 

in whicn cos -J/ may be taken ^vith cither the negative or the 
positive sign; and it is evident that the first will give the 
beginning and the second the end of the eclipse generally. 
For the two interior contacts we have 

Bin 4 = >!!>j}.( -K - ^) 

f~' } (522) 

T = COS 4* ^ cos (M^ — iV ) 


These interior contacts cannot occur when p — Ms less than 
7«o sin {Mq — N), which would give impossible values of sin ^. 

In these formulae we at first assume 2> = 1, and, after finding 
ail approximate value of ^'j we have, by (517), in which x = 0, 
2' =^ M, and in the present problem M= N+ '^i therefore 

^ = JVr+4, (523) 

with which p is found by (518), and the second computation of 
(521) or (522) will then give the required times. We niusst 
employ in (523) the tvvo values of ^^ found by taking cos ij^ with 
the positive and the negative sign ; and therefore different values 
of p wU be found for beginning and ending, so that in the 
second approximation separate computations will be necessary 
tor the two cases. 

In the first approximation the mean values of x', y', and / 
may be used, or those for the middle of the eclipse. With the 
approximate values of r thus found, the true values of r', /, 
and I for the time T= Tq + r may be taken for the secoud 

After finding the corrected value of ^'j we then liave also the 
true value of y = N + '\l/ for each point, and hence also the 
true value of y^ by (518), ^vith which the latitude and longitude 
of the points will be computed by (519). For the local appan*iit 
time of the phenomenon at each place we may take the value 
of ?? in time, which is very nearly the suif s hour angle. 

306. When the interior contacts exist, the rising and Bcttiiiir 
limits form two distinct enclosed curves on the earth's surtaoe. 
If wc denote the times of beginning and ending generally, de- 
termined by (521), by T^ and T^, and the times of interior oon- 
tiict, determined by (522), by y/ and 71', a series of points on 
the risins: limit will be found bv Ait. 303, for a series of time? 
assumed between 7\ and 7^/, and points of the setting limit tor 
times assumed between T^^ and T^, 

When the interior contacts do not exist, the rising and settiii*: 
limits meet and form a single curve extending through the whole 
eclipse. The form of this curve may be compared to that of the 
figure 8 much distorted. A series of points upon it will be 
found by assumiug times between 7\ and T^. 

307. Example. — Let us find the rising and setting limit:* of 
the e-lipse of July 18, 1860, 



Ih'st — To find the beginning and ending on the earth gene • 
rally, we have for the assumed epoch Tq = 2\ page 455, 

m^ Bin M^ = x^ = — 0.081244 
m^ COB M^ = yo = + 0.596075 

which give 

log m^ = 9.77930 
M^ = 352° 14' 19" 

log m^ sin (Jf^ — N) = n9.73938 

n sin -Ar= a/= + 0.5453 
n cos iV= 2^= — 0.1608 


log n = 9.75474 

C08(ifp—iV) = — 0.4336 

For a first approximation, taking 2? = 1, we find, by (521), 

p + l = 1.5367 

log sin ^ 
p + l 


cos ^ = T 

n9 .5528 
m 2».525 


j;__^coB(Jtfp— iVr) = + 2.434 


Approx. beginning Tj 
end r. 



(Jnly 17) 
(July 18) 

Taking cos ij/ negative for beginning and positive for ending, 
we have then, by (518) and (523), 

log tan y 

log Pi 
log tan f 

log sin / 

log sin ^^ 



p + l 

For the above computed times we further find 



200<» 55'.4 

339<» 4'.6 

307 21.2 

85 30.4 





nO. 11605 














log jf 


log n sin N 

log n cos JV 

log n 







106° 23' 50" 

106° 29' 8" 



For a second approximation, therefore, recomputing (521), we 
now find 

and by (618) : 

log sin 4/ 

log cos 4 


log tan / 


200<' 56' 27" 
307 20 17 


339° 4' 68^ 
85 34 6 

Then, for the latitude and longitude of the points, we have. 

by (519), 


— *== 


2r r42" 

357 9 57 

254 38 57 

102 31 

34 38 34 

20*^ 59' 33" 

72 54 8 

91 35 43 

341 18 25 

4 9 46 

Therefore the eclipse begins on the earth generally on July 17, 
23* 54'".5 Greenwich mean time, in west longitude 102® 31' 0" 
and latitude 34° 38' 34", and ends July 18, 4^ 57"*.5 in longitude 
841° 18' 25" and latitude 4° 9' 46". 

It is evident that for practical purposes the first approximation, 
which gives the times within a few seconds, is quite sufficient, 
especially since the effect of refraction has not yet been taken 
into account. (See Art. 327.) 

Secondly, — We now pass to the computation of the cun^e which 
contains all the points where the eclipse begins or ends at sun- 
rise or sunset. In the present example, this curve extends* 
through the whole eclipse, since we have m^ sin {31^ — N) > 1 — /: 
hence the required points will be found for Greenwich times 
assumed between July 17, 23*.91 and July 18, 4*.96. Let us take 
the series 

r, 0*, 0*.2, 0*.4, 0*.6, 0*.8 4».6, 4».8 

The computation being carried on for all the points at once, the 
regular progression of the corresponding numbers for the suc- 
cessive times furnislies at each step a verification of its correct- 
ness. To illustrate the use of the formula*, I give the computa- 
tion for T~ 2*.0 nearly in full. For this time, we find, from 
p. 454 and p. 464, 

X = in sin M= — 0.08124 
y =^ vi cos M = + 0.59608 

I = 0.53675 

rf,= 21o0'49" 
log /Oj = 9.99873 



and hence 

M = 352<' 14' 21" log m = 9.77931 m == 0.60160 

Then, by (517), taking ;? = 1, we have 

ar. CO. log 4tmp 9.61863 

Ij^m—p = 0.13835 log 9.14098 

l^rn+p = 0.93515 log 9.97088 

;i = 26° 49' log 8in» } k 8.73049 

With this first approximate value of X we find the value of p for 
each of the two points, by (518), as follows : 

log tan Y 
log /o, tan Y = log tan / 

O cos r 

loff " — = loff p 

^ cos /' ^^ 


19° 3' 




325° 25 



Repeating (517) with these values of p : 

ar. CO. log 4 mp 
log (I + m^ p) 
log (Z — m + 2?) 

log sin' } X 


M ± X = Y 

log tan Y 

log tan / 

Hence, by (519), 

For T= 2*. (p. 455), fi, 

fi^—^z= at 

Local app. time = ^ in time, 


+ 27° 4' 4" 
19 18 25 


— 27° 0' 26" 
325 13 55 

135° 45' 4" 

28 31 12 

252 46 8 

61 52 35 

9* 3-.0 


242° 36' 45" 
28 31 12 

145 54 27 
50 13 46 
16» 10-.45 

To find whether the eclipse is beginning or ending at these 
points, we have, from p. 465, and by Art. 304, 


14° 17' 

log m sin (M — E) 



\ogp sin (r — E) 





In the same manner are found the results given in the following 
table : 






Long. W. from 

App. Time. 

Mean Time. 





+ 44*^ 


110° 35' 

16* 31-7 


at Sunrise. 




121 33 

15 59 .8 






132 21 

15 28 .7 






144 2 

14 53 .9 






157 6 

14 13 .7 






171 46 

13 27 .0 


" 1 




187 56 

12 34 .4 


a 1 




204 56 

11 38 .3 


Sunset. ' 




221 51 

10 42 .7 






237 54 

9 50 .5 






252 46 

9 3 .0 






266 33 

8 19 .9 






279 17 

7 41 .0 






290 36 

7 7 .7 






300 12 

6 41 .3 


a ■ 

3 .0 



308 12 

6 21 .3 




+ 3 



6 6 .1 




— 3 


320 50 

6 54 .8 






325 53 

5 46 .5 







330 17 

5 41 .0. 






334 4 

5 37 .8 






337 19 

5 36 .8 






340 2 

5 38 .0 






342 9 

5 41 .5 


n ; 


— i 


343 25 

5 48 .4 




+ 25 


99 10 

17 17 .4 


at Sunrise., 




99 33 

17 27 .9 


U I 




101 22 

17 32 .6 






103 52 

17 34 .6 






106 56 
110 34 

17 34 .3 
17 31 .8 



1 .0 






114 50 

17 26 .7 






119 57 

17 18 .3 






126 14 

17 5 .2 





134 15 
145 54 

16 45 .0 


•' '; 




16 10 .5 






168 47 

15 10 .9 






191 43 

13 31 .2 






224 18 

11 32 .9 






249 7 

10 5 .6 





SOLAR ECLIPSE, July 18, 1860.— RISING AND SETTING LIMITS.— (Con/mu^rf.) 

Mean Time. 


Long. W. from 


App. Time. 



+ 62*^ 43' 
58 44 
54 42 
50 35 
46 21 

265*' 37' 
277 27 
286 49 
294 47 
301 53 

9* 11-.6 
8 36 .3 
8 10 .8 
7 51 .0 
7 34 .6 

Ends at Sunset. 

<( ii 
({ ti 






41 55 
37 10 
31 57 
25 55 
18 11 

308 26 
314 40 
320 43 
326 48 
333 18 

7 20 .3 
7 7 .4 
6 55 .2 
6 42 .9 

6 28 .9 

ii ii 
ii (. 
it Ii 
il ii 
ii a 

These points being projected upon a chart (see p. 504), the 
whole curve may be accurately traced through them. It will be 
seen that the method of assuming a series of equidistant times 
gives more points in those portions of the curve where the 
curvature is greatest than in other portions, thus facilitating the 
accurate delineation of the curve. This advantage appears to 
have been overlooked by those who have preferred methods 
(such, for example, as IIansen's) in which a series of equidistant 
latitudes is assumed. 

308. The preceding computations have been made for the 
penumbra ; but we may employ the same method to determine 
the rising and setting limits of total or annular eclipse by 
employing in the formulae the value of I for interior contacts. 
These limits, however, embrace so small a portion of the earth's 
surface that they are practically of little interest 

Curve of Maximum in the Horizon. 

309. To find the curve on which the maximum of the eclipse is seen 
at sunrise or sunset. — When a point of the earth's surface whose 
co-ordinates are f, tj, and ^ is not on the surface of the cone of 
shadow, but at a distance J from the axis of the cone, we have 
the conditions (485), 

J sin ^ = X — ^ 
J cos § = y — 1? 

} (524) 


The amount of obscuration depends upon the distance by 
which the place is immersed within the shadow, that is, upon the 
distance i— J, i being the radius of the shadow on the parallel 
phme at the distance ^ from the principal plane. For the 
maximum of the eclipse, therefore, we have the condition 

(IL ___^-f__Q 

dT dT 

Difterentiating the above equations relatively to the time, and 
denoting the derivatives of a:, ^, &c. by accents, as in Art. 300, 
we have 

^"^ sin Q — Jco8Q.-^ = .r'— r 

dT " " dT 

^^ cos (2 + J sin §. i§- = y'— r/ 

dT ^ ' ^ dT 

which give 

"^"^ = (x!— e') sin Q + (y'— V) cos Q 


The equation L^=l — i^ gives 

and, therefore, 

V— i:'— (a/— ^') sin Q — (/— r/) cos § = (525) 

or, by (510), 

P' = (526) 

This is, therefore, the general condition which characterize:? 
the maximum of the eclipse at a given time. In the j)re!*ent 
problem we have also the condition that the sun is in the horizon, 
for which we may, as in Art. 303, substitute the condition ^, — 0. 
Since, however, the instant of greatest obscuration is not subject 
to any nice observation, a very precise solution of the problem 
is quite unimportant, and we may be satisfied with tlie approxi- 
mate solution obtained by supposing 1^ = 0^ and at the same 
time neglecting the small quantity a' in P'. The condition 
(526) will then be satisfied when in (513) we have 

that is, when 

Q== E or Q = 180* + E 


1, for any given time, the conditions (524) become 

± J sin £ ^ a; - 
± J cos £ = y - 

k with the condition 

t determine the required points of our curve. The angle S 
bre known for tlie given time, being directly obtained from 
Vbulated valuea, but J is unknown. Putting, aa in the 
ndjug problem, 

meia M = 
m cos M = 

p COB}- = 

± d ain E =^ m eia M - 
± J cos E = m COB M - 

afore, putting 4- 

m sin (jtf — E) — p aia(y — E) 
w cos (M — £) — pcoa(jr — E} 

— ^, we have 
nain jM—E) 

± J ^ m COB (Jf — E) — pco6^ 

B first of these equations will ^ve two values of i|/, since we 
IT take cos n^, with the positive or the negative sign ; but, as 
r thoso [tlaces satisfy the problem which are actually icitkiti 
^ bIiuiIow, we must have J < i, or, at least, J not greater than I. 
jit value of il- which would give J> I must, therefore, be 
■ludcd : BO that in general we shall have at a given time but 

^urill be iiuitc accurate enough, considering the degree of 
ion above assigned, to employ in (527) a mean value of p, 
■e p fulls bet^vecn /), and unity, to take log p = ^ log /),. 
if we wiflh a more correct value, wo have only to take 

= * + E 


I then find p as in (518) ; after which (527) must bo recom- 



Having found the true value of ->// by (527), and of y by (528), 
we then have f by the equation 

tan /^ = /»! tan x 

and the latitude and longitude of each point of the curve by (519). 
The limiting times between which the solution is possible will 
be known from the computation of the rising and setting limits, 
in which we have already employed the quantity m sin (wS/— JF); 
and the present curve will be computed only for those times for 
which m sin (Jfef — E) < I, These limiting times are also the same 
as those for the northern and southern limiting curves, which 
will be determined in Art. 313. 

310. The degree of obscuration is usually expressed by the 
fraction of the sun's apparent diameter which is covered by the 
moon's disc. "When the place is so far immersed in the penumbra 
as to be on the edge of the total shadow, the obscuration is total ; 
in this case the distance of the place from the edge of the 
penumbra is equal to the absolute difference of the radii of the 
penumbra and the umbra, that is, to the algebraic sum JL + Lp 
ii denoting the radius of the umbra (which is, by Art. 293, 
negative) ; but in any other case the distance of the place within 
the penumbra is L — d: hence, if D denotes the degree of 
obscuration expressed as a fraction of the sun's apparent 
diameter, we shall have, very nearly. 

This formula may also be used when the eclipse is annular, in 
which case L^ is essentially positive ; and even when J is zero, 
and the eclipse consequently central, the value of D given by 
the formula will be less than unitj', as it should be, since in that 
case there is no total obscuration. 
In the present problem we have 

B ■= ^-^^ (529») 

l + h 

in which I and ?i are the radii of the penumbra and umbra on 
the principal plane, as found by (488). 

Example. — In the eclipse of July 18, 1860, compute the curve 
on which the maximum of the eclipse is seen in the horizon. 



In the computation of the rising and setting limits, the 
quantity m sin {M— E) was less than unity only from 7*= 0*.6 
to r= 4*.2 : so that the present curve may be computed for the 

series of times 0*.6, 0*.8 4*.0, 4*.2. For an approximate 

computation we may take \ogp = Jlog p^= 9.9994, and employ 
only four decimal places in the logarithms throughout 

The computation for 7* = 2* is as follows. For this time we 
have already found (p. 473) 

Hence, by (527), 


log m sin (M — E) 


log sin ^ 

log cos ^ 

logjp cos 4* 

log m cos (3f — E) 

m cos {M — E) 
jpcos ^ 


362<' 14'.4 
14 17.3 
337 57.1 
n9 3538 


+ 0.5575 
+ 0.9727 


Here, if cos o^ were taken with the negative sign we should 
find J= 1.5302, which is greater than I. Taking it, therefore, 
with the positive sign only, we have 

log p^ = 9.9987 

* + E=:r 

log tan z' 
log tan / 

— 13« 4'.3 
+ 1 13. 

with which we find, by (519), 



App. time =: tl^ in time 

176*' 87'.2 
28 31.2 

211 54 

69 1 

11» 46-5 

To express the degree of obscuration according to (529*) we 
have, taking the mean values of I and Ix (p. 454), 

I = 0.5866 
I, = — 0.0092 

i — J = 0.1214 

bJ^^^^ 0.28 

I + I,= 0.5274 0.5274 

In the same manner all the following results are obtained : 





Mean T. 


Long. W. Arom 


App. Local 

Degree of 




+ 24° 44' 
37 47 
47 3 
54 31 
60 38 

107° 41' 
117 47 
127 49 
139 1 
152 24 

17* 19-3 
16 50 .9 
16 22 .8 
15 50 .0 
15 8 .5 


1 .6 
1 .8 

65 20 

68 16 

69 1 
67 34 
64 20 

189 16 
211 54 
233 32 
251 42 

14 14 .1 
18 5 .0 
11 46 .5 
10 31 .9 
9 31 .3 



2 .6 

59 55 
54 41 
48 52 
42 35 
35 49 

266 11 
277 50 
287 31 
295 66 
303 30 

310 S3 
317 22 
324 15 
331 14 

8 45 .3 
8 10 .8 
7 44 .0 
7 22 .4 
7 4 .1 



3 .8 

28 28 

20 21 

+ 11 2 


6 47 .9 
6 32 .6 
6 17 .2 
6 1 .1 


Northern and Southern Limiting Curves. 

811. To find the northern and southern limits of the eclipse on the 
earth's surface. — These limits arc the curves in wliicli are situated 
all the points of the surface of the earth from wliich only a sin»rle 
contact of the discs of the sun and moon can be observed, the 
moon appearing to pass either wholly south or wholly north of 
the sun. They may also be defined as curves to which the out- 
line of the shadow is at all times in contact during it8 progre;?3 
across the earth. 

The solution of this j)roblem is derived from the consideration 
that the sim])le contact is here the maximum of the eclipse, so 
that we must have, as in (526), 

P' = 

and consequently, by (513), 

a' + e sin ((? — JE:) = C/sin (C — F) 



For any given time T^ therefore, we are to find that point of 
the outline of the shadow on the surface of the earth for which 
the value of Q and its corresponding f satisfy this equation. 
This can be effected only indirectly, or by successive approxima- 
tions. For this purpose, we must know at the outset an approxi- 
mate value of Q; and therefore, before proceeding any further, 
we must show how such an approximate value may be found. 

We can readily determine sufficiently narrow limits between 
which Q may be assumed. For this purpose, neglecting a' in 
(530), as well as F^ which are always very small, we have, 

6sin(Q — JB)==C/8in Q 

The extreme values of f are C = ^ and C = !• The first gives 
sin (§ — -&) = 0, and therefore for a first limit we have 

Q=zE or Q = 180° -f E 

The second gives 

6 8in(Q — ^)=/8in Q 



tan ( Q — } JB) = ^-i^ tan } S 

e 4- f 
tan ^ = — ^-^tan iE 

then the equation tan {Q — J-^ = tan o^ gives for our second 

Q=iE+'^ or C = 180° + J^+4' 

To compute o^ readily, put 

tan y = — 

then ' ) (^«1) 

tan 4 = tan (45** + v) tan }-B 

and Q is to be assumed 

between E and i E -{- -^ 

or between 180° + JEand 180° + i E + -^ 
Vol. L— 31 


These limits may be computed in advance for the principal 
hours of the eclipse from the previously tabulated values of 
-E!, e, and /, and an approximate value of Q may then be easily 
inferred for a given time with sufficient precision for a first 

When the shadow passes wholly within the earth, there are 
two limiting curves, northern and southern. For one of these 
Q is to be taken between E and J -E + 4^ 5 for the other, between 
180° + E and 180° + J JS; + o^. Since £J is always an acute angle, 
positive or negative, it follows that when Q is taken between 
U and I E+ '\^jit& cosine is in general positive, while it is nega- 
tive in the other case. The equation jy = y — {I — ?C) ^^ Q 
shows that rj will be less in the first case and greater in the 
second, and hence the values of Q between E and \ E -\- '^ belong 
to the southern limits and the values of Q between 180^ + E and 
180° -\- \ E -\- '^ belong to the northern limit. 

There is only one limit, northern or southern, when one of the 
series of values of Q would give impossible values of Z i*^ ^^ 
computation of the outline of the shadow by Art. 298. But when 
the rising and setting limits have been determined, the question 
of the 'existence of one or both of the northern and southern 
limits is already settled ; for if the rising and setting limits extend 
through the whole eclipse in north latitude, only the southern 
limiting curve of our present problem exists, and vice vert^a; 
while if the rising and setting limits form two distinct cun'Ors 
we have both a northern and southern limiting curve ; and the 
latter must evidently connect the extreme northern and southern 
points respectively of the two enclosed rising and setting curves. 
In our example of the eclipse of July 18, 1860, there exists only 
the southern limiting curve of the present problem, the penum- 
bral shadow passing over and beyond the north pole of the earth. 

Having assumed a value of (?, we find ^^ by the equations (502), 
(504) and (505), and then ^ by (509). This computed value of ^ 
and the assumed value of Q being substituted in (530), this equa- 
tion will be satisfied only when the true value of Q has been 
assumed. To find the correction of §, let us suppose that when 
the equation has been computed logarithmically we find 

log C/sin ((2 — i^) — log [a' + e sin {Q - Ey] =% 
If then dQ and rfj are the corrections which Q and ^ require in 


order to reduce x to zero, we have, by differentiating this equation, 

L a' +e sin (Q — E) J 

dO , dZ 

+ -rz= — ^ 

+ eBin(,Q — E)J A A: 

in which A is the reciprocal of the moduhis of common logarithms. 
In this differential equation we may neglect a' without sensibly 
affecting the rate of approximation. If then we put 

^ dC 

^ ZdQ 

we shall have 

dQ= ^"^ 

cot (§ — JB?) — cot (Q-^F) + g 

This value of dQ is yet to be reduced to seconds by multiplying 
it by cosec 1" or 206266". 

To find ^, we may take, as a sufficiently exact expression for 
computing dQj 



and by differentiating (502) (omitting the factor /t)^, which will 
not sensibly affect g\ 

cos p sin r dp + sin p cos y dy = — I cos Q dQ 
cos p cos Y dp — sin p Any dy z=z I sin Q dQ 

whence, by eliminating dy, 

dp ^ IsiniQ^r) 
dQ co&p 

By (505) a sufficiently exact value of f ^ for our present pur- 
pose is 

Cj = cos p 


d:, . ^dp 

— * = — sm p -^ 

dQ dQ 

g = lsinp sec* p sin (Q — f) (532) 

Putting, finally, 

(? = cot (Q - JB)— cot (§ - F) = - — sin (E -- F) 

^^ ^ v^ y Bin(§ — JB?)sin(Q — iP)"^ ^ 


we have 

aq = f^:«I^" (5a*) 

in which 5.67664 is the logarithm of ^ X 206265". 

When the true value of Q has thus been found, the corre- 
sponding latitude and longitude on the earth's surface are found 
as in Art. 298. 

312. The preceding solution of this problem (which is com- 
monly regarded as one of the most intricate problems in the 
theory of eclipses) is very precise, and the successive approxi- 
mations converge rapidly to the final result. For practical pur- 
poses, however, an extremely precise determination of the limit- 
ing curves of the penumbra is of little importance, since no 
valuable observations are made near these limits. I shall, there- 
fore, now show how the process may be abridged witliout making 
p. ^3 any important sacrifice of accuracy. 

In the first place, it is to be obsened 

that great precision in the angle Q 

.c I is unnecessary. If LM^ Fig. 43, is 

^y^l J the limiting cun^e wliicli is tangent 

at A to the shadow whose axis is at 
^ (7, and if (^ is in error by the quan- 
tity A CA\ the point determined will be (nearly) A' instead of A, 
Xow, altliough yl' may be at some distance from ^1, it is evident 
that it will still be at a proportionally small distance from the 
limiting curve. In fact, we may admit an error of sevvnil 
minutes in the value of Q without scnsibb/ removing the <-omputiMl 
point from the curve. The equation (530), which determines Q, 
may, therefore, without practical error be written under the 
approximate form 


and in this we may employ for ^i the value 

Ci = cos /? 

Hence, having found fl from (502) by employing the first assumed 
value of Qj we then have 

sin (Q — E) / cos ^ 



tan (C — i^) = ^JtZ^^ tan i^ 

e — /cos p 

by which a second and more correct value of Q can be found. 
This equation will be readily computed under the following form : 

tan / = — cos ^5 ) 

« y (535) 

tan (C — \E) = tan (45^ + v') tan iE ) 

The value of Q thus determined may be regarded as final, and 
we may then proceed to compute the latitude and longitude by 
the equations (502) to (508). In this approximate method, loga- 
rithms of four decimal places will be found quite sufficient. 

313. For the computation of a series of points by the preceding 
method, it is necessary first to determine the extreme times 
between which the solution is possible. It is evident that the 
first and last points of the curve are those for which ^j= 0, and, 
consequently, Q = Ej or Q = 180° + E. It is easily seen that 
these points are also the first and last points of the curve of 
maximum in the horizon (Art. 309), and, therefore, the limiting 
times are here the same as for that curve. If, however, we wish 
to determine these limiting times independently (that is, when 
the rising and setting limits have not been previously computed), 
the following approximative process will give them with all the 
precision necessary. 

Since Q = Ej or = 180® + E^ we have, at the required time, 

S = x^lBinE I .^3^. 

rj = y ^^ I cosE ) 

together with the condition (514), for which we may here employ 

e» + iy» = 1 

If we put f = sin fy this condition gives jy = cos f. We have, 
by (512), ^ ^ 

em E = — COB E:= — 

e e 

and we may here regard e as constant. Let the required time 
be denoted by T= T^ + r, T^ being an assumed time near the 
middle of the eclipse. Let 6^,', e/, be the values of 6' and c' for 


the time T^, and denote their hourly changes by 6" and c" ; then 
we have, for the time Ty 

and hence, E^ being the tabulated value of E for the time T^ 

sin ^ = sin ^- H r cos E = cos E^A r 

^ e ^ e 

If, also, Xq, y^y are the values of x and y for the time 7^ a/ and y' 
their hourly changes, we have 

x = x^ + x't y = y^J^tfx 

and the equations (536) become 

sin ;^ = x^ :p i sin ^^ -f I j/ qi — 6" jr 

cosr = yo^ ^cos^o + (y'=P — ^'1^ 

Let m, J!f, n, iV, be determined by the equations 

m sin Jlf = a:^j :+: ? sin E^ 
m cos Mz=zy^zf I cos -£?<> 

tt sin iV = a/ T — ft" ^ (537) 

n cos iV = y q= — c" 

in which the upper sign is to be used for the southern and the 
lower sign for the northern limit ; then, from the equations 

sin y = m sin M -\- n sin N. r 
cos z' = m cos M -{- n cos N . r 
we derive 

sin (/ — N) = m sin (3f — I^T) 

cos (z' — JV) = m cos ( Jf — iV) + nr 

Hence, putting y — iV= t^/, 

sin 4/ = m sin {M — iV) 

^ ^ cos 4. m cos (M — i\r) . 

It is evident that cos i// is to be taken with the negative sign for 
the first point and with the positive sign for the last point of 
the curve. 



To find the latitude and longitude of the extreme points, we 
take Y =N+ '})/, tan y' = p^ tan y, and proceed by (519). 

Example. — To find the southern limit of the eclipse of Jul} 
18, 1860. 

First. To find the extreme times. — Taking T^ = 2*, we have, 
from our tables, pp. 454, 455, and pp. 464, 466, 

Xo = — 0.0812 

y^ = + 0.5961 

I = 0.5367 

E^ = 140 17' 

los: e = 9.7977 

xf= + 0.5452 
y' = — 0.1610 

h" = + 0.0514 
c" = _ 0.0151 

where we take mean values of x', 3/', &c. From these we find 
by (537), taking the upper signs in the formube, 

log m = 9.3555 
log n = 9.7182 

Hence, by (538), 

log sin ( Jf — iV^) = n8.7354 
log sin 4 = n8.0909 
log cos 4= 0.0000 

M = 2890 35' 
i\r=106 28 
Jf— -^=183 7 

log cos {M—'N) = n9.9994 
m cos ( Jlf — N) 


= + 0*.433 

^^=T 1.918 

T= — 1.480 

or T = + 2 .346 

Therefore, for the first and last points of the curve we have, 
respectively, the times 

Tj = 2* — 1».480 = 0».520 
T, = 2 +2.846 = 4.346 

To find the latitude and longitude of the extreme points corre- 
sponding to these times, we have 

log tan Y 
log p^ = 9.9987 log tan / 

Tint Point. 

LMt Point. 

180» 42' 

— 0° 42' 

287 10 

105 46 





21» 1'.4 

20° 59'.8 

6 19.2 

63 42.7 


Henoe, by (519), 



102*> 40' 

339*> 3(K 


16 5 

— 14 47 

Second. To find a series of points on the curve. — ^We begin by 
computing the limits of Q for the hours 0*, 1*, 2*, 3*, 4% 5*. Thus, 
for 0* we have, from the table p. 465, and by (531), 



log tan V 


log tan (45® + v) 

log tan } E 

log tan 4 

2^ 5'.6 
2 16.7 
5® 7'.7 
7 24.4 

For the southern limiting curve, § falls between E and \E-\-^^ 
i.e.j for 0*, between 4° 33' and 7° 24'. In the same manner we 
form the other numbers of the following table : 


Lower limit of Q. 

Upper limit of Q. 


4° 33' 

70 24' 


9 22 

15 18 


14 17 

23 13 


19 14 

30 53 


24 8 

38 4 


28 55 

44 36 

The points of the curve are to be computed for times between 
0*.520 and 4*.346, and we shall, therefore, assume for T the 

series 0*.6, 0*.8, 1*.0 4*.0, 4*. 2, which, with the extreme 

points above computed, will embrace the whole cun^e. 

Instead of determining Q for each of these times by the 
method of Art. 312, it will be sufficient to determine it for the 
hours 1*, 2*, 3*, 4*, and, hence, to infer its values for the inter- 
vening times. ThiL^, for T- 1*, assuming Q — 12®, which is • 



mean between its two limiting values, we proceed by the equa- 
tions (502), for which we can here use 

sin ^ sin /* = X — I sin Q 
sin ^ COS/* = y — I cos Q 

as follows : 

For T^=l\ ( X 

— 0.6266 

log cos p 



+ 0.9170 



Assume Q 


log tan )/ 


a — X — ^ sin Q 



12° 7'.1 

b = y — I cos Q 

+ 0.3920 


4 41.2 

log a — log sin fi sin y 


log tan(45'»+ »') 


log b log sin fi cos y 


tan \E 


log sin fi 


tan(C — J-«^) 


C J-B 

7" 13'.5 


11 54.7 

We thus find, 

for T= 1* 




Q = IP 55', 

22^ 20', 

30" 16', 32" 


From these numbers we obtain by simple interpolation suffi- 
ciently exact values of Q for our whole series of points. And 
since it is plain from Art. 312, that even an error of half a 
degree in Q will not remove the computed point from the true 
curve by any important amount, we may be content to employ 
the following series of values as final : 








































32 .5 

I 1.4 






For each time T we now take x, y, and l^ from the tables of 
the eclipse, and, with the value of Q for the same time, deter- 
mine the required point on the outline of the shadow by the 



complete equations (502) to (508) inclusive, the use of which has 
already been exemplified in Art 302. Employing only four 
decimal places in the logarithms, we shall find that the carve 
may be traced through the points given in the following table : 


Mean Time. 


liong. W. ttom 



+ 16° 5' 

102" 40* 


21 82 

88 81 


25 6 

76 87 


26 36 

69 2 


27 17 

63 9 


27 27 

58 14 


27 15 

53 57 


26 47 

50 9 


26 4 

46 43 


25 9 

43 33 


24 3 

40 34 


22 48 

37 45 


21 5 

34 33 


19 9 

81 25 

3 .2 

16 41 

27 50 


14 14 

24 39 


11 9 

20 44 

3 .8 

8 5 

16 55 


+ 43 

11 46 



5 17 


— 14 47 

839 80 

314. We have applied the preceding method only to the deter- 
mination of the extreme limits of the penumbra, which may be 
designated as the extreme limits of partial eclipse. The same 
method will determine the northern and southern limits of total 
or annular eclipse, by employing the value of / for the total 
shadow — that is, for interior contacts. The latter are, indeed, 
more important, practically, than tlie former, and therefore in 


special cases somewhat greater precision might be desired than 
has been observed in the preceding example. In any such case, 
recourse may be had to the rigorous method of Art. 311. Since 
the limits of total or annular eclipse often include but a very 
narrow belt of the earth's surface, extending nearly equal 
distances north and south of the curve of central eclipse, 
they may be derived, with sufficient accuracy for most purposes, 
from this curve, by a method which will be given in Art. 320. 

The curve upon which any given degree of obscuration can 
be observed may also be computed by the preceding method. It 
is only necessary to substitute J for l, and to give J a value cor- 
responding to D according to the equation (529). All the curves 
thus found begin and end upon the curve of maximum in the 


Curve of Central Eclipse. 

815. To find the curve of central eclipse upon the surface of the 
earth. — This curve contains all those points of the surface of the 
earth through which the axis of the cone of shadow passes. The 
problem becomes the same as that of Art. 298 upon the suppo- 
sition that the shadow is reduced to a point — ^that is, when 
Z — 1^ = 0, and, consequently, by (493), 

S = x 7j =y 

Hence, putting 

the equations (502) to (508) are reduced to the following ex- 
tremely simple ones, which are rigorously exact: 

sin p miy = x 

sin p cos r = 1/1 
c sin C = y^ 
c cos = cos p 
cos f J sin i» = a: ) (539) 

cos f J cos 1* = c cos ((7 + ^1) 
sin f 1 = c sin (C7 -f- ^1) 

tan 0. - 

tan = ^-i— ^ = /J^t — V 

l/(l - ee) 

It will be convenient to prepare the values of y^ for the prin- 
cipal hours of the eclipse ; and then for any given time T taking 
the values of x, i/^ d^, /i^ from the eclipse tables, these equations 
determine a point of the curve. 


316. The extreme times between which the solution is possible, 
or the beginning and end of central eclipse upon the earth, are 
found as follows. At these instants the axis of the shadow 
is tangent to the earth's surface, and the central eclipse is 
observed at sunrise and sunset respectively. Hence, Z being the 
zenith distance of the point Zy we have cos Z=Oy or, by (508), 
^j = 0, whence, by (499), 


which is equivalent to putting sin )9 = 1, or cos )9 = 0, in the 
first two equations of (539), so that we have 

sin Y =^ Xy cos r = yi 

Let a:' and y/ denote the mean hourly changes of z and y^ com- 
puted by the method of Art. 296. Let the required time of 
beginning or ending be denoted by T= Tf^+ r, T^ being an 
arbitrarily assumed epoch ; then, if (x) and (yj) are the values of 
X and 2/1 taken for the time TJ,? we have for the time T, 

sin Y = (.x") + x't 

cos r = 0/0 + y/r 

Let m, M, w, iV, be determined by the equations 

m sin M = (x) n sin N = sf 

m COS M = (yj) n cos iV = y/ 

then, from the equations 

sin Y = VI sm M -\- n sin iV. r 
cos Y = VI cos M -]- n cos iV. t 

we deduce, in the usual manner, 

sin (y — N) ^ VI sin {M — JV) 

cos (j — iV) = m cos (3/ — N) -\- nr 

or, putting 4' = r "" ^> ^^^ solution is 

sin 4 = m sin (M — -AT) 

cos 4. VI cos (M — N) 

n n 

I (540) 



where cos i^ is to be taken with the negative sign for the 
beginning and with the positive sign for the end. 

To find the latitude and longitude of the extreme points cor- 
responding to these times, we have, in (539), cos )9 = 0, sin )9 = 1, 
and, therefore, C= 90°, c = cos y : hence, taking y = iV+ '^j 

COS 9>j sin ^ = sin /* 

cos 9>j cos 1^ = — cos Y sin d^ 

sin 9>j = cos Y cos d^ \ (542) 

tan w. 
tan cp = ^2 — w = fjL. — * 

V^Cl - ee) '^ 

317. To find the duration of total or annular eclipse at any point of 
the curve of central eclipse. — This is readily obtained from numbers 
which occur in the previous computations. Let r=the time 
of central eclipse, t = the duration of total or annular eclipse, 
then T'= T^ J< is the time of beginning or end. Let x and 
y be the moon's co-ordinates for the time T; f and rj those of 
the point on the earth at this time ; x\ y\ f ', jy', the hourly in- 
crements of these quantities ; then, at the time T' we have, by 


(I - iC) sin C = x =F la/^ — (f =p K'O 

(I — iZ) COB Q = y qp iy't — (i^ T } 1^7) 

But we here have a: = f , y = 3^, and we may put ^ = ^^ = cos )9, 

(I ^ i cos fi) sin C = =F (^ — f — 
(I — I cos /5) cos C = =F (y' — 7i') -- 

For the values of f and jy' we have, with sufficient precision, 
since t is very small, 

$' =: fi ( — y &in d + cos ^5 cos d) 
1^'= fj! X sin d 

Hence, by (511) and (512), we find, very nearly, 

a/ — - f ' = </ — f/Qos^ dcos fi = c^ — fcosfi 

y'~V = -y 

If, therefore, we put 

L^l — icoBtS a z:^ (f^f COB fi (648) 


we have 

X8inO = — Xco6Q = — 

2 2 

where we omit the doable sign, since it is only the numerical 

value of t that is required. Hence, we have, for finding tj the 


, ^ a ^ 7200i8inQ ,,,,. 

tan Q = — t = ^ (;M) 

the last equation being multiplied by 3600, so that it now gives 
t in seconds. 

The value of cos j9 is to be taken from the computation of the 
central curve for the ^ven time T, and i, log i, log/, c', V, from 
our eclipse tables. 

318. To find where the central ectipse occurs at noon. — In this case 
we have, evidently, x = 0, and hence, in (539), 

sin fi = y, (545) 

by which ^ is to be found from the value of yj which corresponds 
to the time when x = 0. We then have C= ^, c = 1, tf = 0, 
and therefore the required point is found by the formulae 

9i=fi + d, « = M, (546) 

in which d^ and /i^ are taken for the time when r = 0. 

819. The formulffi (539), (545), and (546) are not only extremely 
simple, but also entirely rigorous, and have this advantage over 
the methods commonly given, that they require no repetition to 
take into account the true figure of the earth. It may be 
observed here that the accurate computation of the central curve 
is of far greater practical importance than that of the limiting 
curves before treated of. 

The formulje (541) must be computed twice if we wish to 
obtain the times of beginning and end with the greatest pos- 
sible precision ; for, these times being unknown, we shall have 
at first to employ the values of x' and y' for the middle of the 
eclipse, and then to take their values for the times obtained by 
the first computation of the formulae. With these new values a 
second computation will give the exact times. 



Example. — To compute the curve of central and total eclipse 
in the eclipse of July 18, 1860. 

It is convenient first to prepare the values of yi = ~ for the 

principal hours of the eclipse, as well as its mean hourly differ- 
ences. With the value log p^= 9.99873 we form, from the values 
of y given in the table p. 464, the following table : 

Gr. T. 




+ 0.91972 

— 0.16095 
















To find the times of beginning and end we may assume 7^= 2* ; 
and for this time we have 

(x ) == m sin Jf = — 0.08124 

a/— n sin i\r— + 0.5453 

(y^) — m cos Jlf — + 0.59782 

y/ — n cos iV^ — — 0.1613 

whence log m — 9.78054 

log n = 9.7548 

M= 352oi5'40" 

N— 106^28'.7 

Employing but four decimal places in the logarithms for a first 
approximation, we find, by (541), 

mco8(J[f — IT) 
cos 4^ 

= + 0*.435 
= =F 1 .468 






- + 




time of 


= 2* 



= 0» 





= 2 


1 .903 . 

= 3 


Taking now x^ and y/ for these times respectively, and re- 
peating the computation^ we have 



y^ = n COS N 



m cos (M — N) 


cos 4 





+ 0.54531 

+ 0.54525 

— 0.16113 

— 0.16164 



106° 27' 42" 

106° 30* 45" 

+ 0».4349 

+ 0».4357 

— 1 .4684 

+ 1 .4685 



213° 23' 12" 

326° 37' 40" 

For the latitude and longitude of the points of beginning and 
end, we now take t' = iV + '\//> and with the values of d^ and 
/ij (pp. 455, 464) for the above computed times, we have 

r = ^+4 

whence, by (542), 



Local App. Time = ^ 


319° 50' 54" 
21 1 15 
13 1 1 


73° 8' 25" 
57 5 3 

45° 36' 50" 
126 3 8 
16* 27"'.9 

15° 45' 34" 
320 53 9 
6* 24".8 

For the series of points on the curve we take the times 1*.0, 

1'^.2, 1*.4 8*.6, 8*.8, which are embraced within the 

extreme times above found, and proceed by (539). Thus, for 
2'^.0 we have 

X = sin /9 sin y 

y^ = sin p cos y 

log sin /9 

log cos /9 = log c cos C 

log y^ = log c sin C 




— 0.08124 
+ 0.59782 





36° 51' 21" 

21 49 



log coos ((7 + ^,) 
log c sin ((7 + dj 

log COS f J sin «9 

log cos f , COS ^ 

log sin f>, 

App. Time = * in time, 

57^ 52' 10" 
351« 17' 13" 
28 31 12 
37 13 59 
57 39 20 
23* 25- 8'.8 

For the duration of totality at this point, we take from pp. 454, 

I = — 0.009082 h'= + 0.1532 

log I = 7.6608 c' = + 0.6011 

log/= 9.3883 

and hence, with log cos ^ = 9.9017 above found, we obtain, by 

i = — 0.012734 « = + 0.4061 

and, by (644), disregarding the negative sign of i, 

t = 211*.3 = 3- 31*.3 

For the place where the central eclipse occurs at noon, we find 
that x == at the time T= 2*. 149, at which time we have 


+ 0.57878 


85» VbZ" 


21 45 


56 1 38 


56 6 57 


80 45 18 

The whole curve may be traced through the points given in the 
following table : 

Vol. 1.-81 






Long. W. Arom 

App. Local 

Duration of 

Mean Time. 










16* 27-.9 






17 21 .3 


' 1'.5 






19 9 .1 








20 26 .6 








21 33 .7 








22 33 .0 








23 25 .1 
















11 .2 








52 .4 








1 29 .8 








2 4 .6 








2 37 .9 








3 10 .9 








3 45 .3 








4 23 .7 








5 15 .1 








6 24 .8 

Northern and Southern Limits of Total or Annular Eclipse, 

820. To find the northern and southern limits of total or annular 
eclipse. — As already remarked in Art. 314, these limits may be 
rigorously determined by the method of Art. 311, by taking 
I = the radius of the umbra {i.e. for interior contacts) ; but I here 
propose to deduce them from the previously computed curve of 
central eclipse. This radius I is assumed to be so small that we 
may neglect its square, which can seldom exceed .0003, and this 
degree of approximation will in the greater number of cases 
suffice to determine points on the limits witliin 2' or 8', which is 
practically quite accurate enough. 

The two limiting curves of total or annular eclipse, then, lie 
so near to the central curve that the value (^^ = cos ^, for a given 
time r, already found in the computation of the latter curve, 
may be used for the former in the approximate equation which 
determines Q. We can, therefore, immediately find Qhj (535), — i.e. 


tan v' = — COS p ) 

e V (547) 

tan (C — \E) = tan (45^ + /) tan }^ ) 

where/, e, and -Bare to be taken from the eclipse tables for the 
lime T. 

The co-ordinates of the point on the central curve correspond- 
ing to the time T being S = x and y^ = 7jy (Art. 315), those for 
a point on the limiting curve may be denoted hy x + dx and 
Vi + ^!/v These being substituted for f and tj^ in the equations 
(499), we have 

dx = — (I — iCj) sin Q ^^i = — G — ^^i) cos Q 

where in the expression for dy^ we omit the divisor /o^, as not 
appreciably changing the value of so small a term. 

Let ^1, t?, 10 be taken from the computation of the central 
curve for the time T, and let f^ + d<p^^ lo + do), be the cor- 
responding values of f^ and (o for the point on the limit for 
the same time. Then, by difterentiating (500), observing that 
rftf = — dof, we have 

cos f J cos ^ rf a* -f- si'^ 9t sin i> d^^ = — dx 
cos f J sin ^ d(o — sin f ^ cos ^ d^^ = — dy^ sin d^ + dZ^ cos d^ 

cos ^^d^^ = dy^ cos d^ -\- dZ^ sin d^ 

whence, by eliminating rff j and substituting f ^ for its value given 
by the third equation of (497), we find 

Ct cos f^dat = — dx (cos f>, cos dj -}- sin ^^ cos ^ sin rfj 

— rf^j sin f ^ sin d 
Cjdf>,= — dx sin 1* sin <fj -|- rfy, cos ti> 

Hence, substituting cos ^ for f j, 

dot = ^ (cos ^ sin § sin rfj + sin d^ cosQ) tan ^ 


, ? — I cos ^ . ^ , 
_| L_ 8in § cos d^ 


Z "^^ t cos B 
df = (sin ^ sin § sin d^ — cos * cos Q) 



These values are yet to be divided by sin 1' to reduce them to 
minutes of arc. It will be convenient to put 

r=— i- t'= 

sin 1' an V 

^^ l^icoQp _ V ., 


cos /9 Bin r cos/9 

in which Z', i', and >l will be expressed in minutes. 

We may in practice substitute df for dy ^ within the limits of 
accuracy we have adopted ; for we find, from the equations on 
p. 457, 

J dtp, cos* w ,1 — ee sin* w 
dip •=. * . = o^pj 

|/(1 — ee) cos'fj |/(1 — ee) 

where the multiplier of dtp^ cannot differ more from unity than 
l/(l — ee) does, — t.e. not more than 0.00835: so that the substitu- 
tion of one for the other ^vill never produce an error of 1' so long 
as dtp^ is less than 5^. 

Finally, adapting the values of doi and dip for logarithmic 
computation, by putting 

A sin -H" r= cos Q 

A cos IT = sin Q sin d^ 
we have ^ (549) 

dm =iX\]i cos (i? — H) tan ip^ + sin Q cos dj 
d^ = Xh sin (t> — H) 

The forraulte (547) give two values of Q diflfering 180°. The 
second value will evidently give the same numerical values of 
dw and d<py but with opposite signs ; and therefore we may com- 
pute the equations (549) with only the acute value of Q^ and then 
the longitude and latitude of a point on one of the limits are 

at -f- dio, ip -\- df 

and those of a point on the other limit are 

w — duty <p — dtp 

The first of these limits will be the northern in the case of 
total eclipse, but the southern in the case of annular eclipse, 
observing always to take I with the negative sign for total eclipse, 
as it comes out by the formulae (487) and (489). 



It is evident that this approximate method is not accurate 
when cos j9 is very small, that is, near the extreme points of the 
curves; and it fails wholly for these points themselves, since 
cos j9 is then zero and the value of X becomes infinite. These 
extreme points, however, are determined directly in a very 
simple manner by the formulse (536), (537), (538), combined with 
(619), by employing in (536) and (537) the value of I for interior 
contacts; and it is with these formulae, therefore, that the com- 
putation of the limits of total or annular eclipse should be com- 

Example. — ^Find the northern and southern limits of total 
eclipse in the eclipse of July 18, 1860. 

I\rsL To find the extreme points. — The values of 6' and c' for 
exterior contacts, from which the values of E on p. 465 are 
derived, differ so little from those for interior contacts that in 
practice, unless extreme precision is required, we may dispense 
with the computation of the latter. For our present example, 
therefore, taking the value of -Efor 7^= 2* and the mean value 
of log 6, as in the computation of the extreme points of the 
southern limit for the penumbra, p. 487, together with 

I — 


we find, by (537), for the northern limit, 

log m = 9.7854 

M — 352*' 


log n = 9.7553 

J\r = 106 


and for the southern limit, 

log m — 9.7731 



log n = 9.7542 

N — 106 


Hence, by (538), 

Northern Limit. 

Sonthem Limit. 


First Point. 

213<> 54'.3 

Last Point. 

326^ 5'.7 

First Point. 

212^ 39^.0 

Last Point. 

327^ 2r.O 

Taking ;• = N+ a//, and the values of d^ and fi^ for these times 
respectively, with log Pi = 9.9987, we find, by (518) and (619), 


log tan / 



320^ 21'.3 

72° 32'.7 

319° 6'.0 

73° 48 .0 







21° 1'.2 

21° O'.O 

21° V2 

21° (K.O 


246 31.7 

96 26.7 

247 26.7 

96 67.7 


13 9.6 

66 64.1 

12 47.1 

67 16.6 


126 37.9 

320 27.4 

126 20.4 

821 18.9 


46 7.7 

16 21.6 

46 2.8 

16 11.4 

Second, To find a series of points between these extremes, by 
the aid of the curve of central eclipse, we assume the same series 
of times as in the computation of that curve, and proceed by 
(547), (548), and (549) ; to illustrate the use of which I add the 
computation for r= 2* in full. From the computation, p. 496, 
we have 

log cos p 

log tan ^^ 


For r= 2* 



361° 17'.2 
21 0.8 
37 14.0 
67 39.3 

Then, by (547), 

(p. 465) log 

log 008 p 

log tan / 


log tan (45° + »/) 
log tan 1 E 
logtan(§— 1J5?) 

Hence, by (549), 


17° 26'.0 
7 8.7 
13° 30'.3 
20 39.0 

By (548), 


log I 

log I' 

log i 

log V 


V sec p 


— 0.009082 


— 39 .16 

— 64.90 

log cos Q == log h sin H 
log sin Q sin d^ = log h cos H 

log h 

82° 18'.2 
268 69.0 



log A 

log A 

log COB (^ — If) 

log tan f>| 

log (1) 

log A 
log sinQ cos <f, 

log (2) 







+ 1'.45 
— 18 .08 

- 16 .63 


log A 

log sin (d—^ 

log d^ 




+ 5r.83 

Hence, for the time r= 2*, we have the two points, 




N. Limit. 

36^ 57.'4 
58 31.1 

S. Limit. 

37^ 30'.6 
56 47.5 

SOLAR ECLIPSE, July 18, 1860. 
Northtm Limit of Total EeUpte. Southern Limit of Total Eel^^se, 




Gr. T. 




46^ 8' 

126*> 38' 


50 18 

116 27 


57 47 

90 57 


60 13 



60 46 

59 40 


60 4 

47 23 


58 31 

36 57 


56 21 

28 9 


53 43 

20 40 


50 43 

14 12 


47 24 

8 44 


43 47 

3 1 


39 49 

357 43 


35 25 

352 6 


30 18 

345 23 


23 31 

335 8 


16 22 

320 27 



Or. T. 




45^ 3' 

125^ 20^ 


50 57 

109 56 


56 45 

87 33 


58 45 

71 46 


59 4 

58 31 


58 19 

47 11 


56 48 

37 31 


54 42 

29 16 


52 11 

22 10 


49 19 

15 56 


46 9 

10 39 


42 41 

5 3 


38 52 

359 51 


34 38 

354 20 

3 .6 

29 45 

347 48 


23 26 

338 20 


15 11 

321 19 


321. The curves above computed j 
lowing chart. 

i all exhibited in the fol- 

Tor the construction of sueh charts, on even a much lai^er 
eealc, the degree of aeeuracy with which our computations 
have been made \a far greater tlian ia necessary, and many 
abridgments may he made which will readily occur to the 
skilful computer.* 

r a grnphic melhod of cam true ting eclipse chnrle, mi k paper bf Hf. 
OBT, ProcmdingB of tha Am. Asaaoiutian for Ihn Adr. of ScieiiM, 8lb 


Prediction of a Solar Eclipse for a Given Place. 

322. To compute the time of the occurrence of a given phase of a 
solar eclipse for a given place. — The given phase is expressed by a 
given value of J, and we are to find the time when this value 
and the co-ordinates of the given place satisfy the conditions 
(485). This can only be done by successive approximations. 

Let it be proposed to find the time of beginning or ending of 
the eclipse at the place. The phase is then J = i — i^, and we 
must satisfy the equations (491). Let T^ be an assumed time, 
and r= Tq-{- r the required time. Let a:, y, x', y\ rf, ?, log i, be 
taken from the eclipse tables (p. 454) for the time T^, Assuming 
that X and y vary uniformly, their values at the time T are 
X + x'r and y + y't. The co-ordinates of the place at the time 
Tq are found by (483) or (483*), in which fx is the sidereal time 
at the place. Putting 

»> = /* — a ^ fi^ — at 

in which at is the west longitude of the place and /i^ may be taken 
from the table (p. 455) for the time T^, the formulse become 

ABmB = pBin^ S = P cos y»' sin * "j 

A cos B = p cos f>' cos * Tj =A sin (B — d) > (550) 

C=^cos(^ — rf) j 

Let f, ^' denote the hourly increments of f Bmdrj; then, assuming 
that these increments also are uniform, the values of the co-ordi- 
nates at the time T are f + f '^ and tj + yfr. The values of f ' 
and yf are found by the formulae (p. 462) 

f = i^ p cos f' cos ^ 

V=A*'f sin d — d'ti 

in which y! and d' are the hourly changes of /i and d multiplied 
by sin 1". The rate of approximation will not be sensibly 
affected by omitting the small term cf'^, and the formulae for f ' 
and yf may then be written as follows : 

r = A*' -A cos * ^ = A*'f sm d (551) 


i = ? — i: 

then, neglecting the variation of this quantity in the first ap- 
proximation, the conditions (491) become, for liie time T, 

i sin § = :r — e + (a/ — e') T 
i cos C = y — iy + (y — ly') T 


Let the auxiliaries m, -Bf, n, and iVbe determined by the eqna- 

mcoQM=y — i) neo8JV=y' — if )^ ^ 

then, from the equations 

iy sin Q = m sin M -\- n sin JV. r 

Z cos Q = m cos ilf -|- n cos JV. r 
we deduce 

i sin (§ — JVO = m sin {M-^N) 

L cos (Q — N) = m cos (Jlf — -AT) -(- nr 
Hence, putting oj' = § — i^T, wo have 

m sin ( Jtf" — If) 
Bin 4 = ^^ ' 


L cos 4 m cos {M — N) 

n n 


by which r is found. Since the first of these equations does not 
determine the sign of cos '^, the latter may be taken with either 
the positive or the negative sign. We thus obtain two values 

of T^= Tq + 7> the first given by the negative sign of 


being the time of beginning, and the second given by the posi- 
tive sign being the time of ending of the eclipse at the place. 

For a second approximation, let each of the computed times 
(or two times nearly equal to them) be taken as the assumed 
time Tq^ and compute the equations (550), (551), (552), (553) for 
beginning and end separately. 

The first approximation may be in error several minutes, but 
the second will always be correct >\'ithin a few seconds, and, 
therefore, quite as accurate as can be required ; for a perfect 
prediction cannot be attained in the present state of the Ephe- 

The formula for r may also be expressed as follows : 

m sin {M — N — 4*) 

n sin 4 

which in the second approximation will be more convenient 
than the former expression ; but when sin n^ is very small it will 
not be so precise. 


If we put 

t = the local mean time of beginning or end, 
we have 

323. The prediction for a given place being made for the 
purpose of preparing to observe the eclipse, it is necessary also 
to know the point of the sun's limb at which the first contact is 
to take place, in order to direct the attention to that point. This 
is given at once by the value of 

which is the angular distance of the point of contact reckoned 
from the north point of the sun's limb towards the east (Art. 295). 

The simplest method of distinguishing the point of contact oii 
the sun's limb is (as Bessel suggested) by a thread in the eye-piece 
of the telescope, ari'anged so that it can be revolved and made 
tangent to the sun's limb at the point. The observer then, by a 
slow motion of the instrument, keeps the limb very nearly in 
contact with the thread until the eclipse begins. The position 
of the thread is indicated by a small graduated circle on the rim 
of the eye-piece, as in the common position micrometer. 

This method is applicable whatever may be the kind of 
mounting of the telescope. Nevertheless, if the instrument is 
arranged with motion in altitude and azimuth, it will be conve- 
nient to know the angle of the point of contact from the vertex 
of the sun's limb, which is that point of the limb which is nearest 
to the zenith. The distance of the vertex from the north point 
of the limb is equal to the parallactic angle which being here 
denoted by y^ is found, according to Art. 15, by the formulae 

p sin y z= cos ^ sin i^ 

p cos z' = sin y» cos d — cos f sin d cos ^ 

(where we have put p for sin !^ and & for the sun's hour angle). 

As Y is not required with very great accuracy, we may here take 

[see (494)] 

pBiny^S p cos r = V 

in which f and rj are the values of the co-ordinates of the place 
at the instant of contact. But, if f and rj denote the values at the 
time T^ we must take 

p Biny = ? + S't pcosr = t^ + 1?'^ (554) 


Ill which we employ the values of f, jy, f ', yf^ and r furnished by 
the last approximation. We then have 

Angular diBtance of the point of contact ftrom 1 = C — f T'Vfi'i^ 

the vertex towards the east, J=iV-}-4 — T 

324. To find the instant of maximum obscuration for a given placCj 
and the degree of obscuration. — At the instant of greatest obscura- 
tion the distance J of the axis of the shadow from the place of 
observation is a minimum.* If we denote the required time by 
Ti = jTq + ^1) the equations of Art. 822 determine r^ for a given 
value of J if we substitute J for L. Denoting the value of 
§ — iV for this case by '^i, we have, therefore, 

J sin 4i = m sin (M — N) 

A cos 4j^ = m cos ( Jtf" — N) + nr^ 

the sum of the squares of which gives 

J« = m« 8in« (Jf — JVO + [m cos (2f — iST) + nr,]' 

Since m and M are computed for the time 7J,, and N is sensibly 
constant, the term m' sin' (M — N) is constant, and therefore J 
is a minimum when the last term is zero, that is, when 

m cos {M — N) 



which quantity is already known from the computation of (553). 
We have, also, 

J == it m sin (^af — JV^ = ± i sin 4, (557) 

in which the sign is to be so taken as to make J positive. The 
degree of obscuration is then given by the formula (Art. 310), 

L + L, 

in which D is expressed in fractional parts of the sun's diameter, 
and L and L^ are the radii of the penumbra and umbra (the 

* More strictly, Z ~ J is a maximum, as in Art. 809 ; but we here >mh||{^^^^ 
small variation of L. The rigorous solution of the problem may be M^- •— ^"^^ 
the condition (526) P' = ; but the above approximation is sufficfen^ " 


latter being negative) for the place of observation. From (488) 
we find, by putting 8ec/= 1, 

and hence 

L — A 

D = (658) 

in which k = 0.2723. 

If we neglect the augmentation of the moon's diameter, or, 
which is equivalent, the small difference between L and 2, and 


we have ' ^ ^ 

2> == e qp e sin 4» 

where the lower sign is to be used when sin '^ is negative, so 
that D is always the numerical difference of e and e sin a//. Li this 
form e may be computed for the eclipse generally, and n^ will be 
derived from the computation for the penumbra for the given 
place. A preference should be given to the value of a// found 
from the computation for the time nearest to that of greatest 
obscuration, which is usually that used in the first approximation 
of Art. 322. 

Example. — ^Find the time of beginning and end, &c., of the 
eclipse of July 18, 1860, at Cambridge, Mass. 
The latitude and longitude are 

f> = 42° 22' 49" Id = 71° 7' 25" 

For this latitude we find, by the aid of Table III., or by the 
formulse (87), 

log p sin tp' = 9.82644 log p cos / = 9.86912 

With the aid of the chart, p. 504, we estimate the time of the 
middle of the eclipse at Cambridge to be not far from 1*. Hence, 
taking 7J, = 1* for our first approximation, wo take for this time, 
from the eclipse tables, p. 454, 

z r= — 0.6266 ^=-\- 0.6468 I = 0.6868 

y = -f 0. 7667 y* = — 0. 1606 log i = 7.66287 

rf= 200 67'.4 ^1= 180 8r.2 log /*'= 9.41799 



Hence, by (550) and (551), 

fii—o — '» — 


B = 

59 24.6 



V — 




^ = 

— 0.0686 

and, by (552) and ( 


mBin M z — ^ 

— 0.0020 

m cos Jf y — 7 

+ 0.2728 



M = 

8590 84'.7 

M—N — 

256 84.1 

log sin 4 


log 008 4 -= 


log^= 9.7858 

t; = 0.0028 

/; = / — if= 0.5340 

n sin ^'= z*— ^'=r + 0.4416 
n cos iV= y'— tf = — 0.1020 
log n = 9.G562 
JV= 103«0'.0 

m cos (Jf — K) 

L cos 4. 

+ 0*.140 

_ 1.028 

_ f — 0.888 
^~"lor+ 1.168 

Approximate time of beginning = 0^.117 

end = 2 .163 



Taking then for a second approximation T^ = 0*.12 for begin- 
ning, and Tq= 2^.16 for end, we shall find* 












+ 0.00601 

+ 0.89783 

+ 0.57034 

4- 0.54528 

+ 0.54530 


— 0.16090 

20° 57' 45" 

20° 56' 53" 

19 8 

30 55 13 



289° 11' 43" 

319° 47' 48" 


— 0.47755 

+ 0.53915 

+ 0.42423 



+ 0.06368 

+ 0.14793 



* The values of 7f and y' here employed are not those given in the table p. 446, 
but their actual values for the time T^^ as given in the table of zf and y' on p. 




m Bin M 

m Qo^M 

log m 


n sin N 

n cos JV 

log n 














-- 0.40774 

+ 0.48356 

+ 0.35868 

+ 0.14611 



311^ 20' 16" 

73° 11' 15" 

+ 0.48160 

+ 0.39737 

-- 0.09471 

— 0.11620 



101^ r 32" 

106*^ 18' 0" 

210 12 44 

326 53 15 

210 44 

328 49 56 

— 31' 16" 

— 1° 56' 41" 

+ 0*.0197 

+ 0*.0800 


2 .2400 

0* 8-23- 

2* 14-» 24- 

4 44 30 

4 44 30 

19 23 53 

21 29 54 

July 17. 

July 17. 

Sir 51' 32" 

75*^ 7' 56" 

Local time, 

Angle of Pt.of Contact from \ 
North Pt. of the sun = V 

A third approximation, commencing with the last computed 
times, changes them by only a fraction of a second. 

To find the angular distance of the point of contact from the 
vertex of the sun's limb, we have from the second approximation, 
by (564) and (555), 

f + f 't = ;) sin r 
Tj -^ Tj'r =p cos ^ 


Angle from vertex = Q — y 


— 0.6974 
+ 0.5379 
307° 38'.8 
4 12.7 


-- 0.4658 
+ 0.4206 
312° 4'.5 
123 3.4 

The time of greatest obscumtion is best found from the first 
approximation, which gives, by (656), 


T = 1*. 

_mcoB(3f-J\r)^' ^^^^ 

n * ' 

Tj = 1M40 

= 1* 8* 24'. 

w = 4 44 80 

Local time of max. obscur. = t = 20 23 54 

For the amount of greatest obscuration we have, also, from 
the first approximation, by (667) and (568), 

i = 0.6340 logi= 9.7275 

k = 0.2723 log sin 4 == n9.6955 

Z^k =0.2617 logJ=: 9.4230 

2(i — A:) = 0.5234 J= 0.2649 

i — J= 0.2691 


Or, by (559), taking as constant the value of e found by employ- 
ing the mean value I = 0.5367, i.e. 

e = 1.015 
we have 

« gin 4 = — 0.503 

D = 0.512 
which is quite accurate enough. 

326. Prediction for a givai place by the method of the American 
Uphemeris. — This method is based upon a transformation of 
Bessel's formula suggested by T. Henry Safford, Jr., and, with 
the aid of the extended tables in the Ephemeris, is somewhat 
more convenient than the preceding. The fundamental equa- 
tion (490) gives, by transposition, 

(x — ?)*=(/-: tan/)« — (y — 7))' 
the second member of which may be resolved into the factors 

6=(Z_:tan/) + (y-i?) 

c = (Z — C tan/) — (y—ri) 
or, by (494), 

b = 1 -{- y — /> sin ^' (cos d -{- am d tan/) 

+ p cos (p' (sin d — cos d tan /) cos ^ 

c = 1 — y -f- /> sin ^' (cos d — sin rf tan /) 

— P cos y' (sin d + cos d tan /) cos * 


If we put 

A=x B = l + y C= — l + y 

E = coad + Bin d tan/ = cos (d — /) sec/ 
F = COB d — Bin d tan/= cob (d +/) sec/ 
G=i Bin d — cos d tan/ = sin (d — /) sec/ 
H=Bin d + cos d tan/ = sin (d + /) sec/ 

all of which are independent of the place of observation and are 
given in the Ephemeris for each solar eclipse, for successive 
times at the Washington meridian, we shall then have to com- 
pute for the place 

a = X — ( = A — p cos ^ sin ^ ^ 

6 = -B— -^ /» sin ^+ G^ /» cos ^ cos * I (560) 

c = — G -\- F pBin^ — JEp cos ^' cos d ) 

and the fundamental equation becomes 

We have here, as before, t? = /i^ — a; ; and the value of fi^ is 
also given in the Ephemeris for the Washington meridian. 

If now for any assumed time T^ we take from the Ephemeris 
the values of these auxiliaries, and, after computing a, 6, and c 
by (560), find that a differs from >/6(?, the assumed time requires 
to be corrected ; and the correction is found by the following 
process. Put 

m = i/bc, 
o^, y, m'= the changes of a, b, m, in one second, 

r = the required correction of the assumed time ; 

then at the time of beginning or ending of the eclipse we must 

a + o't = m -f m'r 

m — a 

a' — m' 

To find a' we have, by differentiating the value of a and de- 
noting the derivatives by accents, 

a'= A' -^fi'p cos ^ cos * (661) 

Vol. L— 83 


in which fJ denotes the change of //^ in one second, and is the 
same as the fJ of our former method divided by 3600. 

To find ml we have, following the same notation, and neglect- 
ing the small changes of E^ F^ G-j JS, 2, and/, 

V = B' ^ /i' Gp cos ^ sin d^ 
d = — C + plRp cos f>' sin * 

Since / is small, we may in these approximate expressions pot 
G = Hj and hence 

V= — d= B'— fi! Gp cos f^ sin^ (561*> 

Now, from the formula m* = 6c, we derive 

2 mm' = d>'+ h(f= (c — 6)^ 

which, if we assume 


m'= — ycotQ 

and therefore r is found by the formula 

m — a 

a' + 6' cot Q 


The Ephemeris gives also the values of A', 5', and C, which 
are the changes of -4, J5, and C in one second. These changes 
being very small, the unit adopted in expressing them is .000001; 
so that the above value of r, as also the value of /£' in (561), 
must be multiplied by 10^ The formulse (560-568) then agree 
with those given in the explanation appended to the Ephemeris. 

It is easily seen that Q here denotes the same angle as in the 
preceding articles ; for we have at the instant of contact 

tan Q = = = 

m! h — c y — ly 

Examples of the application of this method are given in every 
volume of the American Ephemeris. 



826. The preceding articles embrace all that is important in 
relation to the prediction of solar eclipses. Since absolute rigor 
is not required in mere predictions, I have thus far said nothing 
of the effect of refraction, which, though extremely small, must 
be treated of before we proceed to the application of observed 
eclipses, where the greatest possible degree of precision is to be 


327. That the refraction varies for bodies at different distances 
from the earth has already been noticed in Art. 106 ; but the 
difference is so small that it is disregarded in all problems in 
which the absolute position of a single body is considered. 
Here, however, where two points at very different distances from 
the earth are observed in apparent contact, it is worth while to 
inquire how far the difference in question may affect our results. 

Let SMDA, Fig. 44, be the path 
of the ray of light from the sun's 
limb to the observer at A^ which 
touches the moon's limb at M; SMB 
the straight line which coincides with 
this path between 5^ and M^ but when 
produced intersects the vertical line 
of the observer in B, It is evident 
that the observer at A sees an ap- 
parent contact of the limbs at the 
instant when an observer at B would 
see a true contact if there were no 
refiraction. Hence, if we substitute 
the point B for the point A in the 
formula of the eclipse, we shall fully take into account the effect 
of refraction. 

For the purpose of determining the position of the point By 
whose distance from A is very small, it will suffice to regard the 
earth as a sphere with the radius p = CA, It is one of the pro- 
perties of the path of a ray of light in the atmosphere that the 
product qfx sin i is constant (Art. 108), q denoting the normal to 
any infinitesimal stratum of the atmosphere at the point in which 
the ray intersects the stratum, fx the index of refraction of that 
stratum, and i the angle which the ray makes with the normal. 


If, then, /t>, [jL^y Z' denote the values of 3, //, and i for the point -4, 
we have, as in the equation (149), 

qix Bin I = p[i^ sin Z' 

in which Z' is the apparent zenith distance of the point M^ and 
/1q is the index of refraction of the air at the observer. 

Now, let us consider the normal q to be drawn to a point D of 
the ray where the refractive power of the air is zero, that is, to 
a point in the rectilinear portion of the path where /i = 1. Then 
our equation becomes 

q sin i = pix^ sin Z' 

in which q = CD, i = MDF = CDB. Putting Z = the true 
zenith distance oi M= MBVj and s = the height of B above 
the surface of the earth = ABy the triangle CDB gives 

(p + s) BinZ = q sin i 

which with the preceding equation gives 

- 5 Au sin Z' ,^, 

In order to substitute the point B for the point A in our com- 
putation of an eclipse, we have only to write p + s for p in the 

equations (483), or /> 1 1 H — J for />. Therefore, when we have 

computed the values of log f, log ij, and log !^ by those equa- 
tions in their present form, we shall merely have to correct them 

by adding to each the value of log ( 1 H I This logarithm 

may be computed by (564) for a mean value of /Ji^ (= 1.OOO2800) 
and for given values of Z. For Z we may take the true zenith 
distance of the point Z (Art. 289), determined by a and d. But 
by the last equation of (483) we have so nearly cos Z= ^ that 
in the table computed by (564) we may make log ^ the argu- 
ment, as in the following table, which I have deduced from that 
of Bessel {Astron. Untersuchungen, Vol. II. p. 240). 




Correction of logs, 
of ^ J7, C- 











Correction of logs, 
of S, n, f. 









The numbers in this table correspond to that state of the at- 
mosphere for which the refraction table (Table 11.) is computed ; 
that is, for the case in which the factors ^ and y of that table are 
each = 1. For any other case the tabular logarithm is to be 
varied in proportion to ^ and r- 

It is e\ident from this table that the eflTect of refraction will 
mostly be very small, for so long as the zenith distance of the 
moon is less than 70° we have log ^ > 9.53, and the tabular 
correction less than .000001. From the zenith distance 70° to 
90° the correction increases rapidly, and should not be neglected. 



828. If 5' is the height of the observer above the level of the 
sea, it is only necessary to put p + s^ for /> in the general formulae 
of the eclipse ; and this will be accomplished by adding to log $, 

log ly, and log C the value of log 1 1 H — I, which is {M being 

the modulus of common logarithms) 



But s' is always so small in comparison with p that we mav 


neglect all but the first term of this formula ; and hence, by 
taking a mean value of p (for latitude 45°) and supposing ^ to 
be expressed in English feet^ we find 

Correction of log e, log 17, log C = 0.00000002079 8^ (565) 

For example, if the point of observation is 1000 feet above 
the level of the sea, we must increase the logarithms of f, 3;, 
and C by 0.0000208. 

If s' is expressed in metres^ the correction becomes 0.000000064 y. 


329. To find the longitude of a place from the observation of an 

eclipse of the sxcn. — The observation gives simply the local times 

of the contacts of the disc of the sun and moon : m the case of 

partial eclipse, two exterior contacts only ; in the case of total or 

annular eclipse, also two interior contacts. 


w =r the west longitude of the place, 

t = the local mean time of an observed contact, 

fi = the corresponding local sidereal time. 

The conversion of t into /jl requires an approximate knowledge 
of the longitude, which we may always suppose the observer to 
possess, at least with suflicient precision for this purpose. 

Let Tq be the adopted epoch from which the values of z and y 
are computed (Art. 296), and let 

^o» Vo = *^® values of x and y at the time T^y 

a/, 1/ = tlieir mean hourly changes for the time t -\- w; 

then, if we also put 

r = t + w-'T^ (566) 

the values of x and y at the time t +to (which is the time at the 
first meridian when the contact was observed) are 

The values of x' and y' to be employed in these expressions 
may be taken for the time t + to obtained by employing the 


approximate value of oi, and will be sufficiently precise unless 
the longitude is very greatly in error. 

The quantities I and i change so slowly that their values 
taken for the approximate time t + co will not differ sensibly 
firom the true ones. For the same reason, the quantities a and d 
taken for this time will be sufficiently precise : so that, the latitude 
being given, the co-ordinates f , 7, ^ of the place of observation 
may be correctly found by the formulce (483). Since, then, at 
the instant of contact the equation (490) or (491) must be exactly 
satisfied, we have, putting L=l — if. 

2/ sin Q = x^ — ^ -{- afr 
X cos Q = y^ — 1? + y'r 

} (567) 

in which r is the only unknown quantity. Let the auxiliaries 
niy My n, N be determined by the equations 

m%m M=zx^ — ? n sin JV:= j/ 

mco8Jf=y„ — 17 ncosiV=y' 

then, from the equations 

i sin C = wi 8i^ -3f + n sin JV . r 
L cos Q = m cos Jtf" -(- n cosiV. r 

by putting 4^=6 — iV, we obtain 

m sin (M — N) 

sm 4 = -z 

X cos 4* mooB(itf' — N) 

n n 

m sin {M — N — 4) 
n sin 4 

} (568) 


where the second form for r will be the more convenient except 

when sin 4^ is very small. As in the similar formulae (553), the 

angle '^ must be so taken that L cos 4^ shall be negative for 

first contacts and positive for last contacts, remembering that in 

the case of total eclipse X is a negative quantity. 

Having found r, the longitude becomes known by (566), which 


m = To — ^ + r (570) 


If the observed local time is sidereal, let fi^ be the Bidereal 
time at the first meridian, corresponding to 7^; then, r being 
reduced to sidereal seconds, we shall have 

and this process will be free from the theoretical inaccnracy 
arising from employing an approximate longitude in converting 
fx into L 

The unit of r in (569) is one mean hour ; but, if we write 

hLcos^ hmcos^M — J\r) 

n n 

- m sin (ilf — JV — 4») 

= A : i 

n sm^* 

we shall find r in mean or sidereal seconds, according as we take 
h = 3600, or A = 3609.856. 

330. The rule given in the preceding article for determining 
the sign of cos '\^ (which is that usually given by writers on thw 
subject) is not without exception in theory, although in practice 
it will be applicable in all cases where the observations are 
suitable for finding the longitude with precision ; and, were an 
exceptional case to occur in practice, a knowledge of the approxi- 
mate longitude would remove all doubt as to the sign of the term 

— . But it is is easy to deduce the mathematical condition 

for this case. 
At the instant of contact, the quantity 

is equal to L^. At the next following instant, when r becomei» 
r + (h, it is less or greater than L^ according as the eclipse is 
beginning or ending. If then we regard i* as sensibly constant^ 
the differential coeflicieut of this quantity relatively to the time 
must be negative for first and positive for last contacts. The 
half of this coefficient is 

(X, - c + x^r) (a/- e') + (y, - ly + y'r) (y'- ,0 

(where the derivatives of f and tj are denoted by c' and ij'), or, by 
(567), putting N + ^i^ for Q, 

L [sin {X + 4) (^ - c ') + cos (.V + 4) (^ - V)] 


Computing f ' and yf by the formulae (551), or, in this case, by 
^'= fi'p cos f ' cos (ji — a) rj'=: //( sin d 

and putting 

n' sin N'= of— f' n'cos N'= y'— V 

the above expression becomes 

in'cos(JV— i\r'+4) 

Hence, when L is positive, that is, for exterior contacts and 
interior contacts in annular eclipse, 4^ must be so taken that 
cos(iV— iV' + 4^) ^^^^^ he negative for first and positive for last 
contact. That is, for first contact 4^ ^^^ be taken between 
N' — N+ 90"" and N'—N+ 270''; and for last contact between 
If'—N+ 90° and N'—N— 90°. For total eclipse, invert these 

In Art. 322, we have N = iV', and hence the rule given for 
the case there considered is always correct. 

331. To investigate the correction of the longitude found from an 
observed solar eclipse^ for errors in the elements of the computation. 

AX, Ay, Aiy = the corrections of x, y, and L, respectively, 

for errors of the Ephomoris, 
A?, Aiy = the corrections of S and rj for errors in p and f»', 
At = the resulting correction of r. 

The relation between these corrections, supposing them very 
small, will be obtained by differentiating the values of L sin Q 
and L cos Q of the preceding article, by which we obtain 

Aiy BinQ -\- L cosQ^Q = ax — a$ + a/ at 
Ai cos Q — iy sin § A § = Ay — Aiy -f y AT 

where ax and Ay, being taken to denote the corrections of 
X = x^,+ x'r andy = yo+ y'^, include the corrections of x' and y'. 
Substituting in these equations n sin N for x' and n cos N for 
y', and eliminating a§, we find 

aL = (ax — A$) sin Q + (Ay — Aiy) cos g + n cos (g — JV) . at 

and substituting for Q its value iV+ i^, 

. . sin (iV+ 4*) , .C08(JV+4') . ^i 
AT = — . (ax — Af ) ^ • — - — (Ay — Aiy) - — 5^ '—^ -I 

n COS 4^ n cos 4 n cos 4» 



1 1 

AT = ( Ao? sin iV + Ay COS -AT) H — ( — aje; cos iV' -f Ay sin N) tan 4 

1 1 

-| — (a? sin JV4- Aiy cos iV) (— a? cosi\r4- A17 sin JV^ tan 4 

/• fw 

+ ^^ (571) 

which is at once the correction of r and of the longitude, since 
we have, by (570), aa; = at. 

832. In this expression for at, the corrections ax, Ay, 4c. have 
particular values belonging to the given instant of observation 
or to the given place. In order to render it available for deter- 
mining the corrections of the original elements of computation, 
we must endeavor to reduce it to a function of quantities which 
are constant during the whole eclipse and independent of the 
place of observation. For this purpose, let us first consider 
those parts of at which involve ax and Ay. For any time 7^, at 
the first meridian, we have 

y =y.+ n COB NiT.^T,) 

X sin JV -|- y cos JV = x,, sin iV^ + y„ cos JV4- n ( T^ — T^ 
— X cos iV + y sin JV = — Xq cos iV + y^ sin iV 

The last of these expressions, being independent of the time, is 
constant. If we denote it by x ; that is, put 

X = — Xq cos iV -f y^ sin JV= — x cos N + y sin N (572) 

we shall find from the two expressions 

xx + yy = x>c + [x^sin JVr+ y.cos iVr+ n(7;— T.)]' (573) 

This equation shows that the quantity |/xx + yy, which is the 
distance of the axis of the shadow from the centre of the earth, 
can never be less than the constant x, and it attains this minimum 
value when the second term vanishes, that is, when N + y^cos N + n(^T,^ T.) = 
and hence when 

T, = T, — 1 (Xosin N+ y„cos iV^ (574) 


which formula, therefore, gives the time 7\ of nearest approach of 
the axis of the shadow to the centre of the earth, while (572) 
gives the value of the distance of the axis from the centre of the 
earth at this time. By the introduction of the auxiliary quanti- 
ties 7\ and x, we can express the corrections involving ax and Ay 
in their simplest form ; for we have now, for the time of obser- 
vation < + a>, 

a: sin JV + y cos J\r:= x^ sin iV -f y^cos iV -|- » (^ -f a> — T^ 

= n (t + a» — T,) 

and if An, aTj and ax are the corrections of w, 2\, and x on 
account of errors in the elements, we have 

AX sin JV+ Ay cos N= — n aT^ + (^ + a» — 2^) An 1 
— AX cos iVT 4- Ay sin JV^ = AX / ^^ ' ^^ 

These expressions reduce those parts of at which involve ax and 
Ay to functions of a Tj, am, and ax, which may be regarded as 
constant quantities for the same eclipse. 

We proceed to consider those parts of at which involve a^ 
and ^yj. These corrections we shall regard as depending only 
upon the correction of the eccentricity of the terrestrial meridian ; 
for the latitude itself may always be supposed to be correct, 
since it is easily obtained with all the precision required for the 
calculation of an eclipse ; the values of a and d depend chiefly 
on the sun's place, which we assume to be correctly given in the 
Ephemeris ; and [i is derived directly from observation. Now, 
we have (Art 82), e being the eccentricity of the meridian, 

cos cp . , (1 — ee) sin w 

l/(l — ee sm' ^ ) l/ (1 — ee sm' ip) 

whence, by diflferentiation, 

A/ocosy^ _ , pp sin* y^ 

= p cos • 

Aee '^ 2(1 — ee)« 

A./psiny^ . , pp sin' y^ p sin y^ 

= P Sm • r — 

Aee ^ 2(1 — eey 1 — ee 

or, putting 

P sin ^' 

/5 = 

1 — ee 


A./9 COS / 

A . /> sin ip* 
From the values 

= } ^^p cos 9/ 

= ipfipAnf/'^fi 


S = p cos ^' sin (;ei — a) 

19 = /> sin f ' cos 4 — p cos 9/ sin (2 cos (;* — a) 

we deduce 

and hence 

A^sin N+ A7C08iV=}^^( f sin iV + 9 C08 iV') Am — )3 eos cf cos i\r Am 
— A^ cos iV -f- A7 sin iV = J j3/3 ( — ^ cob H+if sin iV) Am — fi cos (i sin i\r Aee 

The values of ^ and 37 may be put under the forms 

f = x^— (x^— f) = rTo— m sin ilf 

^ = yo — (yo— 'y) = yo— wi cos jf 

by which the second members of the preceding expressions are 
changed respectively into 

i P^ {. ^0 ^^^ ^ + y©^®* ^ — "* ^®* (^ — ^)^ ^^^ — P cos d COS A*" Am 
and i pp [— ZqCOS N -\- y^Bin N -^ m sin (i/ — iV)] Am — ^ cos d sin JV Am 

or, by (574) and (572), into 

i PP [n (TJ, — 7\) — TO cos (M— iV)] Am — P cob d cos iV Am 
and \ PP\_ X -\- mBia (M — iV)] Am — /? cos <2 sin iV Am 

or, again, by (569) and (570), into 

^ Pp\n(t -\-ti— T^) — L cos 4*] Am — P cob d cos iV Am 
and h PP\. X -\- L Bin -^l Am — p cos d sin iV Am 

Hence, that part of at which depends upon ^ee is equal to 

PP r f* I m\ ^ , r nx 5 COS d COS (iV -f 4) 

^ [n (« 4- w — 7\) — X tan "4/ — /i sec 4] Am — t:^ ^ !— ^ Am 

2n n cos 4* 

When these substitutions are made in (571), we have 

Ar = Aw = Aa7\ -f A tan 4 . ^ — A (< 4- 0) — Tj) — + A sec 4/ . — 

n n « 

+ - Fi-^/? [» (« + " - r.) - » tan 4 - Z sec 4] - ^£2L^i^iH'±±n A« (577) 

W L C03 4 J 


where we have multiplied by h to reduce to seconds. The unit 
is either one second of mean or one second of sidereal time, 
according as r is in mean or sidereal time. If the former, we 
take h = 3600; if the latter, h = 8610. 

333. The transformations of the preceding article have led us 
to an expression in which the corrections a 7\, ax, ati, and ^ee are 
all constants for the earth generally, and which, therefore, have 
the same values in all the equations of condition formed from 
the observations in various places. But a still further transform- 
ation is necessary if we wish the equation to express the rela- 
tion between the longitude and the corrections of the Ephemeris, 
so that we may finally be enabled not only to correct the longi- 
tudes, but also the Ephemeris. 

Since a7\, ax, aw are constant for the whole eclipse, we can 
determine them for any assumed time, as the time 7\ itself. For 
this time we have 

x sin JV + y cos N=0 
— a:cosiV4- y8iniV = x 
AX sin iV + Ay cos N= — n a2\ ^ (^^^) 

— AX cos iV + Ay sin JV = ax 

The general values of x and y (482) may be thus expressed : 


^ = ^T;r- y 

sm TT "^ smn 


X= cos ^ sin (o — a) F=: sin dcosd — cos ^ sin d cos (o — a) 

From these we deduce 

aX att aF A?r 

^^ = ^t;;^ - ^ T^;rz ^y = ^i:rz-y 

smir tauTT ^ sinff ^ tanw 


Az sm iV 4- Ay COS iv = ! (« Bin JV + y cos JV) 

sin ff tan ir 


-AX cosiV+ Ay sin JV= ~^^cosiV+AFsiniV ^^ cos JV- y sin N) 

sinff ^^ ^ ^tanff 

and for the time T^ these become, according to (578), 

^rn A-tTsin JV+ aFcos JV 
* sm n 

— A J!" COS N+ A Fsin N a^ 
AX = . — ! X 

sm T tansr 


A . p COS <p' 

A . p sin ^' 

From the values 

= } pfip cos f ' 

= }i9;9/9 8insp'--i9 


^ = p cos f»' sin (;ei — a) 

iy = /t) sin f ' cos d — p cos 9/ sin d cos (;* — a) 

we deduce 

^=}^^e -^=}^/?iy-/9cos<f 

and hence 

Af 8iiiiV+ A^C08iV=}^^( f 8iiiiV+ J7C08iV) A^e — )3 cos </ cos A^ A«r 
— Af C08iV-f- A^sin iV= J ^^ (—^ cos iV+ 17 sin iST) Aee — /? cos <2 sin .V Am 

The values of f and 7 may be put under the forms 

$ = ar^,— (^o~ = ^0— ^^ sin 3f 

^ = yo — (yo — ^) = yo — ^ cos jtf" 

by which the second members of the preceding expressions are 
changed respectively into 

J P? [ ^Q^^^ N" 4- yo<i09 N — m 008 (J/ — iV)] Ar? — (i cos */ cos .V A^f 
and J /?/?[— Zq cos iV -f- yosin iV -f- m sin (i/ — iV)] Aw — ,? cos d sin ->' A« 

or, by (574) and (572), into 

} PP \n {T^ — Tj) — m cos (M — N)^ Arr — /? cos d cos X A« 
and }/?/?[ X -f m sin (AT — iV)] A« — p coa d sin A" Af« 

or, again, by (569) and (570), into 

i Pl^ [n {t + u— T^) — L cos 4] Am — /? cos rf cos A' Afif 

and i ft? [ X -}- /y sin 4] ^^'^ — /? co8 ^^ sin A'' Aw 

Ilence, that part of at wliich depends upon ^ce is equal to 

Pt^ r /* , m\ * , r , -i a /? COS </ COS (A' -f «^> ^ 

^ [n (t + u — TJ) — X tan 4/ — /^ sec 4/] Aw — ^^ ^^ l_I- ^(f 

2n n COS ^ 

When these substitutions are made in (571), we have 

Ar = Aw = Aa7\ + A tan 4/ . ^ — A (< + o — r,) — -f A sec 4 . — 

fi n n 

, ^Fioor /- I /n\ 1 • r .t /3 COS rf COS (A' - -^ )"| . 

-f - J /?;?[»(< + w — 7\) — X tan 4 — Jy sec 4] — ^- i ^ I Aw ^oi • ) 

LOXGIirDK. '^-"^ 

where we have multiplied bvA to reduce x? fe: r:.^!-. Tlo \:v.:: 
is either one second of mean or onr >ec":'i:'i oi ?::oro;r: v.'.r.o. 
according as r is in mean or sidereal r.ii.e. K tLe iori:»«.r, wo 
take h = 3600; it* the laner, k = ot.lO. 

333. The transformations of the preoed:nir article luwo Ud v.s 
to an expression in which the correot:oL> aT. ax. a;i, aiul ai aiv 
all constants for the eanh ^enerallv. and which. thcr*.'tori\ luwo 
the same values in all the equations oi condition lorniod tVviu 
the ohservations in various places. But a still further transtoini- 
atiou is necessarj- if we wish the e4uation to express tlio rela- 
tion between the longitude and the corrections of the Kphonieris, 
80 that we may finally be enabled not only to correct tlio louiri- 
tudes, but also the Ephemeris. 

Since aT,, ax, a?i are constant for the whole eclipse, wo can 
determine them for any assumed time, as the time 7; itself. Kor 
this time we have 

X sin X + y cos iT = 
XQOSX+ ysin y = x 
Aj sin T 4- Ay cos JV = — n A r 

-a:co8.V+ ysin y = x \ .. , 

— AJcosJV-p Ay sin jV= AX ) 

The general values of x and y (482) may be thus exprcssevl: 

sinw ^""Bin:^ 


^= cos ^ sin (a - a) 7= sin a cos d - cos 5 sin d cos ^« '-^ 
Prom these we deduce 

A-*! A;r a ir v- 

AX = r - ^' "•• 

whence ""^^ ^^''' ^y = ^->'u.ur: 

sin TT 
Bin T 

— Azco8.V + AyBin^':=,:3A£co8iV4-Ar8miV .. ■.. V) . 

^j + (x COS N y »*'" '^ !#» ' 

and for the time T Watx^ v 

^itJiese become, according to (57H), 

-nAT,=^£^n^+ aYcos .V 

sm TT 

Bin 7 tiiii 



Again, by differentiating the values of X and F, we have 

A JT := COB ^ cos (o — o) A(a — ol) — Bin ^ sin (a — a) A* 
A Y = [cos ^ cos d '\- WDih Bin d cos (a — a)] a^ 

— [sin ^ sin (Z + cos ^ cos d cos (a — a)] a4 
+ cos d sin (2 sin (a — a) A(a — a) 

But for the time of nearest approach we may take a = a ani) 
put cos (5 — d) = 1, whence 

A-r= cosd.A(o ^-a) aF= a(^ — d) 

so that 

_ sin i? cos d . A(a — a) + cos iV . A(d — d) 
— nA i ,== : 

sin iz 

: (679) 

— oos-ZVcos ^. A(o — a) -|- sin JV. A(^ — d) Ax 

Ax == --— ^-^---— ——-----— —--y-— ---——— ————— M -►^— 

sin TT tan -k 

To find An, which depends upon the corrections of 7f and y, 

we observe that x' and y', regarded as derivatives of x and y, are 

of the form 

^ dX \ , dY \ 

dT sin^r dT sinx 

But -j=^ and -^^ depend upon the changes of the moon's right 

ascension and declination, which for the brief duration of an 
eclipse are correctly given in the Ephemeris. The errors of t! 
and y\ therefore, depend upon those of;:: so that if we write 

sm n sm TT 

and regard a and h as correct, we find 

Ax'=~a/ ^}/ '= — y 

tauTT tauTT 

From the equations n sin iV= a:', n cos iV= y', we have 


An sin iV4- n AiVcos JV= Aa/= — nsin iV 

An cos JV— n AiVsin J\r=rAy'= — ncosJV 




whence, by eliminating ^N^* 

:^ = ^^ (580) 

» tan n 

Since A(a — a), ^{d -— rf), A;r will in practice be expressed in 
seconds of arc, we should substitute for them a (a — a) sin 1", 
a(* — d) sin 1", A;r sin 1'' in the above expressions ; but if we at 
the same time put it sin 1" for sin tt and tan «■, the factor sin 1" 
will disappear. 

To develop aZt, we may neglect the error of the small term i ^ 
and assume aX = a?. We have from (486) and (488), by 

neglecting the small term k sin tt^ and putting ^ = 1, z = -: y 

the following approximate expression for I : 

, sin JJ , 

1 = -— — ±k 


which gives 

Ai == aZ = -^ =b A* -« — . — (681) 

Substituting the values of aT^i, ax, ah, and a{ given by (579), 
(580), and (581), m (577), and putting 

the formula becomes, finally, 

Au=. — y[ sin JVoo8dLA(a — a) -|- eo8 JV.A(<5 — d)} 

+ V [— 008 iVeos d.A(a — a) -|- sin y,^{6 — d)] tan 4 

+ v| —;- =tZ IT A* I 800 4 


+ v|ii (< 4- « — 7\) — » taii4 — -j-boo^Iait 

+ v[}i5^[i»(< + «-r,)-»tan + -2i8004]-^^5li^^!^ 


where the negative sign of «:aA: is to be used for interior contacts. 
It is easily seen that n^k represents very nearly the correction 

* The angle iVis independent of erron in ir, iinoe tan iVcs >: so that we might 
haTe taken AiV=0. 


of the moon's apparent semidiameter, and —j- that of the son'g 

semidiameter ; and that nt^e is the correction of the assumed 
reduction of the parallax for the latitude 90°. 

334. Discussion of the equations of coTuUtion for the ccrrectim of 
the longitude and of the elements of the computation. — The longitude 
w found by the equation (570), (Art 329), requires the correction 
^Q) of (582). If, for brevity, we put 

y = Bin N cos d A(a — «) + COS JVa(^ — d) 1 (^ag\ 
^ = — cos iV COS d A(o — a) + sin N^(d — d) i 


ctf'= the true longitude, 

we have the equation of condition 

ai' = 01 -f Aai = Ai — v/* + y tan 4 . ^ -|- &c. (584) 

If the eclipse has been observed at several places, we can form 
as many such equations as there are contacts observed. If the 
observations are complete at all the places, we can, for the most 
part, eliminate from these equations the unknown corrections of 
the elements, and determine the relative longitudes of the sevenl 
places ; and if the absolute longitude of one of the places is 
known, that of each place will also be determined. 

I shall at first consider only the terms involving y and i?. The 
quantity v;' is a constant for all the places of observation, and 
combines with w, so that it cannot be determined unless the 
longitude of at least one of the places is known. If then we put 

Q = at' -\- vy a = V tan 4/ 

the equations of condition will assume the form 

fl _ rt,9 _ a» = 

Suppose, for the sake of completeness, that the four contacts 
of a total or annular eclipse have been observed at any one place, 
and that the values of the longitude found from the several con- 
tacts by Art. 329 are w^, w^, cw,, o)^. We then have the four equa- 

[1] fl — a^ 1^ — a»j = 

[2] fl — a, * — o», = 
[8] C — a, 1^ — «, = 
[4] C — a^i^ — «^ = 


where the ntimerals may be assumed to express the order in 
which the contacts are observed ; [1] and [4] being exterior, and 
[2] and [3] interior. In a partial eclipse we should have but the 
1st and 4th of these equations. 

Since exterior contacts cannot (in most cases) be observed with 
as much precision as interior ones, let us assign different weights 
to the observations, and denote them by je>i, p^y p^j p^, respectively. 
Combining the four equations according to the method of least 
squares, we form the two normal equations 

[p ] fl — . [pa ] »» — Ipw ] = 
Ipa] Q — Ipaa] & — [pao*] = 

where the rectangular brackets are used as symbols of smnma- 
tion. From these, by eliminating £, and putting 

we find 

P,> -f Q = (585) 

from which the value of t> would be determined with the weight 
P. But the computation of Q under this form is inconvenient. 
By developing the quantities P and Qj observing that [j>aa] = 
Pfii + P^ + P^ + PfiiJ ^^'j we shall find 

P— PiPi {^i — «»)* + Pi Ps (^i — ^'a)' -h Pi Pi («i — «4)* 

Pi -\- Pt -{- Ps -\- Pi 

I Pi P% (g» — «»)* -f Pt Pi {^i — g*)' + Ps Pi (Oj — ^4)* 

Pi+Pt-^-Pz +Pa 

Q ^ P\ Pt (<'i — «») ('■^i — "2) + Pi Ps (fli — gg) ("1— s) -f Pi Pi (gj — Oj) K — s) 

Pi -i-Pi-^Ps+Pi 

, PtPt (g» — 09) (S — S) + Pt Pi i^i — «<) (^2 — "4) + PsPi (^5 -- ^4) (S - ^4) 


These forms show that if we subtract each of the equations [1], 
[2], [3] from each of those that follow it in the group, whereby 
we obtain the six equations 

(aj — a,) 1? -f oij — 01, = 
(«i — a,) * + a*j — «/, = 
(aj — a J iJ + CI*, — ci*^ = 

(«. — «») t> + cii^ — 013 = 
(«« — aj ^ + a*. — 01^ = 
(d. — aj !?+.«;, — 01^ = 
Vol. I.— 34 


and combine these six equations according to the method of 
least squares, taking their weights to be respectively 

VxP. P.P. ,4^ 

Vi+p» + ;>a +i>4 Pi+p»+ Pt+P4, 

we shall arrive at the same final equation (585) as by the direct 
process, with the advantage of avoiding the multiplication of the 
large numbers w^^ a;,, &c. 

Suppose that at another place but three contacts have been 
observed, the true longitude being w''^ and the computed longi- 
tudes oij, w^j fi>y, and that, having put i2'=ctf"+ vy^ we have 
formed the three equations 

[5] Q! — a, i> — fl'ft = with tlic weight p^ 
[6] iy— .dji* — tt*j = " " p, 

[7] fi'— ay*— flly = " " pj 

The subtraction of each of the first two from those which follow 
gives the three equations 

(«5 — «6) * + "'s — "'e = 
(a* — a,) * + fi*, — "'y = ^ 

(«e — ^) ^ + fi'e — "'j = ^ 

of which the weights will be respectively, according to the above 


PsP^ Pi Pi PtPi 

and the combination of these three equations, according to 
weights, will give a normal equation of the form 

which gives a value of # with the weight P'. 

Now, suppose that this method applied to all the observations 
at all the places has given us the series of equations in i?, 

P»!> + Q =0 
P tJ + Q' = 
P".^+ Q"=0,&c.; 

then, since P, P', P", &c. are the weights of these several deter- 
minations, the final normal equation for determining tf, derived 
from all the observations, is 


that is, it is simply the sum of all the individual equations in & 
formed for the places severally. 

The same reasoning is applicable to any of the terms which 
follow the term in t? in (584) ; so that if we suppose all the terms 
to be retained, this process gives an equation in t? for each place, 
in which besides the term P9 there will be terms in aA:, ^H, &c., 
and from all the equations, by addition, a final normal equation 
(still called the equation in d) as before. In the same manner, 
final normal equations in aA:, a1?, &c. will be formed. Thus we 
shall obtain five normal equations involving the five unknown 
quantities t?, aA:, aJJ, A;r, Ae6, which are then determined by 
solving the equations in the usual manner. But, unless the 
eclipse has been observed at places widely distant in longitude, 
it will not be possible to determine satisfactorily the value of 
^jtj much less that of Me. It will be advisable to retain these 
terms in our equations, however, in order to show what effect an 
error in tt or ee may produce upon the resulting longitudes. 

When I?, &c. have been found, we find J2, £', &c. from the 

equations [1], [2] . . . . [5], [6] The final value of S will be 

the mean of its values [1 — 4] taken with regard to the weights ; 
and so of £', &c. Hence we shall know the several differences 
of longitude 


— a,"= Q-^Qf, a,' — a,'"= fi — fl", &c. 

K one of the longitudes, as for instance (o'y is previously 
known, we have 

and hence all the longitudes become known. 

Finally, from the values of y and t> the corrections of the 
Ephemeris in right ascension and declination are obtained by 
the formulfle 

cos ^ A(a — a) = sin iV. y — cos N , ^ 1 r5S6> 

A(5 — (f) = C08iV.;'+ sin iV.d I ^ 

335. When only two places of observation are considered, one 
of which is known, it will be sufficiently accurate to deduce y 
and t> from the observations at the known place (disregarding 
the other corrections), and to employ their values in finding the 
lou^tude of the other place. 



336. TVTien good meridian observations of the moon are avail- 
able, taken near the time of the eclipse, tlie quantities A(a — a\ 
a(5 — d) [for which we may take A(a — a'), a(5 — 5')], may be 
found from them. The terms in y and «> may then be directly 
computed by (583) and applied to the computed longitude ; after 
which the discussion of the equations of conditiou may with 
advantage be extended to the remaining terms. 

337. Before proceeding to give an example of the computation 
by the preceding method, it will be well to recapitulate the 
necessary formulse, and to give the equations of condition a 
practical form. 

I. The general elements of the eclipse, a, d, /, log i, ar, y, x', y, 
are supposed to have been computed and tabulated as in Art. 297. 

n. The latitude of the place being ^, the logarithms of p cos f ' 
and p sin f' are found by the aid of our Table HI., or by the 
fomiulie (87). 

The mean local time t of an observed contact being given, 
find the corresponding local sidereal time p ; also the time ^ 4- « 
at the first meridian, employing the approximate value of the 
longitude (o, 

[If the observed time is the sidereal time /i, the time /£ -f « at 
the fii^st meridian, converted into moan time, will give the 
approximate vahic of t + to,'] 

For the time t -{■ (o take a, d, Z, and log i from the eolipe 
tables, and compute the co-ordinates of the place and the radium 
of the shadow by the formulte 

A i^\n B = p sin 9"' 

A cos B ^^ p cos f'cos (fjL — a) 

^ =P cos f ' sin (fi — a) 
jj = A sin {B — d) 
Z =Acos{B — d) 

AVhon log ^ is small, add to log ^, log 3j, and log ^ the correc- 
tion for refraction, from the table on p. 517. 

III. For the assumed epoch 7i at the first meridian (being the 
epoch from which the mean hourly changes jr' and y' are reck- 
oned), take the values of x and y from the eclipse tables, 
denoting them by 0*0 and y^. Also the mean hourly changed x* 


and y' for the time t + o}. Compute the auxiliaries m, M, &c. 
by the formulse* 

mmi M =x^ — f n sin i\r = a/ 

m cos M= y^ — 17 n cos N = y' 

m sin fJlf — N) 
sm <4' = ^^ 

where 1^ is G'^ general) to be so taken that L cos ij/ shall be 

negative for a first and positive for a last contact (but in certain 

exceptional cases of rare occurrence sec Art. 330). 


hL cos 4 Am cos (M — N) 

n n 

or, when sin ^ is not very small, 

hm sin(Jtf" — N — 4) 

^ sin 4 

If the local mean time t was observed, take h = 3600 in these 
formulae, and then the (uncorrected) longitude is found by the 

oi = 7; - f + r 

If the local sidereal time /i was observed, take h = 3609.856, 
in the preceding formulae ; then, /i^ being the sidereal time at the 
first meridian corresponding to TJ, we have 


= /*o — /* + ^ 

The longitudes thus found will be the true ones only when 
all the elements of the computation are correct 

IV. To form the equations of condition for the correction of 
these longitudes, when the eclipse has been observed at a suffi- 
cient number of places, compute the time 7J of nearest approach, 
and the minimum distance x, by the formulae 

2; = 7; - 1 (X, sin N+y, cos N) 

X = — x^ COS i\r + y© ^^^ -^ 

* The yalues of JV and log n being nearly constant, it will be expedient, where 
many obserrations are to be reduced, to compute them for the several integral hours 
at the first meridian, and to deduce their yalues for any given time by simple 


Take k for the time 7\, and compute the logarithm of 

the same value of h heing used here as before. 
For each observation at each place compute the coefficients 

V tan 4') ^ sec ^^^ &ud 

E = vnCt +at — T.) — XV tan 4. r- »* sec 4 

where the unit of / + a; — T^i is one mean hour, 

r» 1 OiO r /J I m\ * • r n Vi3 COS rf 001 (A' -1- 4) 

F= ipp [vn(< -f « — 7\) — XV tan <4' — 2^ v sec <4'] i 2I 

000 4 

in which 

J = 959".788 log H = 2.98218 

^ ^ ^osin^' J _ ^ 9.99709 

1 — ee 

Then, a>' denoting the true longitude, the equation of condition is 

L>'= (J — v.y + v tan 4'*v=zzy8ec4'*^Ait-|-v8ec<4'* h -i?Ajr -f /*. t A« 

where the negative sign of the term v6ec4'«^AA: is to be useil 
for interior contacts. 

The discussion of the equations thus formed may then be 
carried out by Art. 334; taking as the unknown quantities 

Y, t?, TT^k, —J- J A'T, and TZMe, 

Example. — ^Find the longitude of Washington from the fol- 
lowing observations of the solar eclipse of July 28, 1851 : 

At Washington (partial eclipse) : 

Beginning of eclipse, July 27, 19* 21- 31v2 M.T. 

End " « " 20 50 38.0 " 

At Konigsberg (total eclipse) : • 

Beginning of eclipse, July 28, 3 38 10 .8 « 

Beginning of total obsc, '' ** 4 38 57 .6 " 

End of total obscuration, " " 4 41 54 .2 " 

End of eclipse, " " 5 38 32 .9 " 



For these places we have given — 

Lai. ^ 

Washington, + 88^ 53' 39".25 
Konigsberg, +54 42 50 .4 

Long, o 
+ 5» 8- 11'.2 
— 1 22 0.4 

The longitudes are reckoned from Greenwich. That of 
Konigsberg will be assumed as correct, while that of Washington 
will be regarded as an approximate value which it is proposed 
to correct by these observations. 

I. The mean Greenwich time of conjunction of the sun and 
moon in right ascension being, July 28, 2* 21* 2'.6, the general 
eclipse tables will be constructed for the Greenwich hours 0*, 1*, 
2*, 3*, 4*, and 5* of July 28. For these times we find the follow- 
ing quantities from the Nautical Almanac : 

For the Moon.* 

Qreenwich mekn 




July 28, 0» 

125*^40' 6".75 

+ 20^ 3'30".00 

eC 27".30 


126 19 9 .41 

19 58 9 .86 

28 .41 


126 58 10 .80 

19 52 39 .99 

29 .49 


127 37 10 .82 

19 47 1 .92 

30 .54 


128 16 9 .37 

19 41 15 .21 

31 .56 


128 55 6 .36 

19 35 19 .89 

32 .56 

For the Sun. 

Greenwioh mean 




July 28, 0» 

127° 6' 5".25 

4- 19° 5' 24".70 



8 32 .63 

4 50 .28 



10 59 .99 

4 15 .74 



13 27 .34 

3 41 .21 



15 54 .67 

3 6 .64 



18 21 .99 

2 32 .05 


* The moon's a and 6 in the Naut. Aim. are directly computed only for every noon 
and midnight and interpolated for each hour. I have not used these interpolated 
Talues, but have interpolated anew to fifth differences. The moon's parallax has 
been diminished by (y\Z according to Mr. Adams's Table in the Appendix to the 
liaut. Aim, for 1856. 



"Witli these values we form the following tables, as in Art 297 : 




KxterioT Oontects. 

Interior CoBtarta. 





127® 6' 17".22 

19® 6' 16".66 



— 0.011771 1 7.661131 


8 89 .61 

4 42 .76 




82 1 


11 1 .78 

4 8 .96 






13 24 .03 

8 86 .14 






16 46 .27 

8 1 .80 






18 8 .60 

2 27 .46 













+ 67 


— 66 









— 0.769366 

— 0.199776 
+ 0.869816 
+ 0.939860 
+ 1.608766 

+ 0.669684 

+ 0.968689 

— 0.083020 


J- 9 



Hence the mean changes r' and y\ for the epoch 7i = 2* (ac- 
cording to the method of Art. 296), and the corresponding values 
of N and log n, are as follows : 






+ 0.569563 


98° 18' 39".7 





19 42 .7 




20 45 .3 





21 47 .5 





22 50 .0 





23 52 .7 


II. Tlie full computation for Konigsborg, where both exterior 
and interior contacts were obsorv^ed, will ser\'e to illustrate the 
use of the preceding formula^ in every practical case. 

For ^ = 54° 42' 50".4 we find 

log p Bin sp' = 9.909898 

log p cos f ' = 9.762639 

The sidereal time at Greenwich mean noon, July 28, was 
8* 22"* 13\27, with which /i is found as given below. The com- 
putation of c, jy, and L will be as follows : 




lit Sxt. Cont. 

lit Int. Cont. 

2d Int. Cont. 

2a Ext. Cont. 


8* 88- IC.S 

4* 88« 57'.6 

4* 41- 54'.2 

5* 38"« 32'.9 

t + u 

2 16 10.4 

8 16 57.2 

8 19 53.8 

4 16 32.5 


12 46.44 

18 1 43.22 

18 4 40.81 

14 1 28.31 

fi (in arc) 

180® 11' 86".6 

195® 25' 48". 8 

I960 10' 4".7 

210«22' 4".7 

For < + «, a 

127 11 40 .1 

127 14 4.2 

127 14 11 .2 

127 16 25 .6 

" d 

19 8 59 .8 

19 8 25 .6 

19 3 28 .9 

19 2 52 .0 


52 59 56 .5 

68 11 44 .1 

68 55 53 .5 

83 5 39 .1 

log sin (fi — a) 





log C08 (/I — a) 











+ 0.462862 

+ 0.587528 

+ 0.540244 

-f 0.574748 

log Asm B 





log ^ 008 ^ 






66<» 47' 82".2 

75« 10' 40".4 

75<>38' 5".9 

85* 6'14".3 

B — d 

47 48 82 .4 

56 7 14 .8 

56 34 42 .0 

66 8 22 .3 

log A 





log sin (5 — d) 





log COS (^ — d) 











+ 0.654289 

+ 0.697888 

+ 0.700152 

+ 0.745427 






' < 4- «, log i 





" / 

-f 0.538956 



-f- 0.533772 


+ 0.002789 

H- 0.002148 

-f 0.002117 

4- 0.001524 


+ 0.531217 



-f 0.532248 

in. The epoch of the table of x' and y' being 7^= 2*, we have 
for this time 

x„ = — 0.199775 

y, = + 0.802185 

with which we proceed to find the values of w. 

m sin if — Xq — ^ 




— 0.774523 

« cos -¥ — yo — 7 

4- 0.147946 

-f 0.104297 

+ 0.102033 

4- 0.056758 

log m sin M 

n9. 820948 




log m cos M 






282« 35' 42". 8 

278<> 3' 5".4 

277«51' 1".5 

274« 11'28".3 

log m 





Por < + «, N 

98<>21' 2".l 

98«22' 6".l 

98«22' 8".2 

98<>23' 7".3 

" " log n 







179<»41' 0".3 

179* 28' 53".3 

176« 48' 21".0 

log sin (M — .V) 

n8. 869321 







log Bin 4 

log8in(if— A^— 4) 

A = 3600, log A 



186« 26' 27".7 
368 49 13 .0 
4- 0* 16- 24^.0 



343« 1' 8".6 

196 39 61 .7 


-f 1* 10-« 12'.0 

118.148016 i 
fi9.681958 j 
208O 44' 14^.0 

6<> 7'Sr.2 
330 44 39 .8 1G9 40 47 .8 





+ 1*19* S-.i; 4-2*15-o2'5 

— 1 38 10.8 —2 38 57.6 —2 41 64.2 —3 38 32.9 

— 1 22 46.8 —1 22 45.6! — 1 22 46. ij —1 22 40.4 

IV. JSquations of condition. — To find T^ and x, we have for 

log x^ = n9.3006 

logyo= 9.9043 

log n = 9.7602 

x^ siniV 

l/o COS N 


= + 0.3434 

= + 0.2023 
T; = 2*.5457 

7r= 3630" 

— x. cos JNr= — 0.0290 log J =2.9822 

+ y^BinN= + 0.7938 
X = + 0.7648 

log X = 9.8835 


log — 

= 0.0066 
= 3.5599 

= 9.4157 

log fi = log ^-^^ = 9.9128 

1 — ee 

log y = log -^ = 0.2362 


With these constants prepared, we readily fomi the cooflBcients 
of the equations of condition as follows : 

l8t Ext. Cont. 


1st Int. Cont. 


2d Int. Cont. 

i adKzt.Cunt. 

log tan 4 






log sec 4 


' 0.0194 



V tan 4 

-f 0.163 

' —0.526 

+ 0.944 

-f 0.185 

V sec 4 


i -f- 1.801 

— 1.964 

-r 1.738 



-f 0*.7355 

-f 0».7860 

-: l*.73n<) 

log (ti-u-T,) 





vn{t^u- 7\) 


-f 0.7295 

-i- 0.7795 

-^ 1.7155 

— XV tan 4 

- 0.1251 

-f 0.4023 

— 8.7228 : 

— 0.1414 


V sec a 


-r 0.4506 

— 0.4691 


— 0.4512 


-^ 0.0516 

-i 0.6627 

-u 0.5689 ' 

- 1.1229 



— XV tan 4 

— Lv 8604 


log let pari of F 

log cos (iV + 4) 
log ( — vp cos d sec 4) 

log 2d part of F 
Ist part of F 



•• Jf 

Ut Sxt Oont. 

Itt Int. Cont. 


2d Ext. Oont 

— 0.1261 
+ 0.9192 

-f 0.7295 
+ 0.4028 
+ 0.0264 

4- 0.7796 

— 0.7228 

— 0.0276 

-f 1.7166 

— 0.1414 

— 0.9222 

4- 0.6202 

+ 1.1672 

-1- 0.0296 

-f 0.6619 

288* 46'.2 


81*» 2r.8 



807« 4'.9 




104» 28'.8 



-f 0.1741 
-f 0.8184 

+ 0.8878 
— 0.2092 

+ 0.0099 
+ 0.9160 

+ 0.2182 
-f 0.8849 

-f 0.4926 

H- 0.1781 

+ 0.9269 

+ 0.6581 

Putting a»' + v;* = J2, we have, therefore, for the four Kouigs- 
berg observations, the equations 






0=— 1* 22«46«.8 + 0.168 1^ — 1.730 TTA*— 1.780 — +0.062 A7r + 0.498ffAw 

0=— 1 22 46.6—0.626 —1.801 +1.801 +0.668 +0.178 
= — 1 22 46.1+0.944 +1.964 —1.964 +0.669 +0.926 
0=— 1 22 40.4 + 0.186 +1.788 +1.788 +1.128 +0.668 

where we have annexed a column for the weight p, giving 
interior contacts double weight. 

A similar computation for the two observations at Washington 
gives the following equations, in which S'=^ w" + vj^ cw" de- 
noting the true longitude of Washington : 



0* = 6» 7-« 29».9 + 1.660 ^ — 2.8927rA* — 2.892 — — 2.681 Att + 0.722 n-Af* 
0^=6 7 21.9 — 2.406 +2.969 +2.959 +0.609 —1.828 

More observations would be necessary in order to determuic 
all the corrections ; but I shall retain all the terms in order to 
illustrate the general method. Subtracting each of the Konigs- 
berg equations from each of those which follow it, we obtain the 
six equations, 








= 4- 1'.2 — 0.689 & — 0.071 tA* + 3.531 — + O.Cll Ajt — 0.315tAm 

= + .7 -f 0.781 + 8.694 — 0.284 + 0.617 -f 0.483 

= -f 6 .4 + 0.022 -f 8.468 + 8.463 + 1.071 -f 0.060 

= — .6 4- 1.470 + 3.765 — 8.765 — 0.094 + 0.748 

= 4-5.2 + 0.711 4-8.534 —0.068 4-0.460 4-0.375 

= -f 5 .7 — 0.759 — 0.281 4- 8.697 4- 0.554 — 0.373 

where the weight in each case is the quotient obtained hy 
dividing the product of the two weights of the equations whose 
difterence is taken, by the sum of the weights of the four 
original equations (Art 334). 

The same method, applied in the case of the two Washington 
equations, gives the shigle equation 



= — S'.O — 4.066 i» 4- 5.851 ttA* 4- 5.851 — 4- 8.190 At — 2.055 tAi« 

From the equations (A') and (B') are formed the following 
final equations, having regard to their weights, in the usual 
manner : 

= 4- 15.495 4- 10.426 1» — 5.300 tA* — 16.377 — — 6.609 At 4- 5.281 tA«i 

= — 12.445 — 5.300 4- 84.506 4- 6.185 4- 10.040 — 2.575 

0..= — 8.191 — 16.377 4- 6.135 4-84.505 4-10.740 —8.214 

--: — 9.371 — 6.G09 4- 10.040 4- 10.740 4- 5.672 — 8.316 

^ -f 7.951 4- 5.281 — 2.575 — 8.214 — 3.816 4- 2.675 

As we cannot expect a satisfactory determination of at an<l ta^v 
from these observations, we disregard the last two eqiiatioiiN 


and then, solving the first three, we obtain «?, jtaA:, and — - in 
terms of ah- and .tacy, as follows : 

»> :^ — 4".36 4- 0.375 AT — 0.525 rj^ee 
taA: = 4- .02 — 0.216 at — 0.004 ta^« 

^= — 1 .83 — 0.095 AT — 0.010 TAt« 

These values substituted in the equations (A) give 

Q^^V 22- 44*.38 4- 0.651 at 4- 0.432 ta^^ 
fi == — 1 22 46 .64 4- 0.684 4- 0.443 
fi = — 1 22 46 .58 4- 0.685 4- 0.442 
Q=^ — l 22 44 .34 4- 0.653 4- 0.432 


the mean of which, giving the second and third double weight, is 

(A") fi = — 1* 22" 45'.86 + 0.674 att + 0.439 r dee 

The equations (B) become 

fl'= 5» 7- 26-.99 — 1.314 att — 0.116 r^ee 
fi'=6 7 27.03 — 1.314 —0.101 

the mean of which is 

(B") ^1= 5» 7- 27'.01 — 1.314 att — 0.109 r^ee 

Now, if we assume the longitude of Konigsberg to be well 
determined, we have 

Q = ai'+ yr = — 1* 22- 0*.4 + vy 

which, with the equation (A''), gives 

vr = ^ 46«.46 + 0.674 ajt + 0.439 xA^e 

Hence, by (B ''), the true longitude of Washington is 

!»"== C— vy = 5» 8- 12'.47 — 1.988 ajt — 0.548 n^^ee 

If the longitude of Washington were also previously well estab- 
lished, this last equation would give us a condition for deter- 
mining the correction of the moon's parallax. Thus, if we adopt 
fi,''=5* 8"* 12'.34, which results from the U.S. Coast Survey 
Chronometric Expeditions of 1849, *50, *51, and '55, this equation 

= + 0.13 — 1.988 ATT — 0.548 rAec 

Ar == 4- 0".07 — 0.276 rAee 

The probable value of A^e, according to Bessel, is within 
dz 0.0001, so that the last term cannot here exceed 0".10. If, 
therefore, the above observations are reliable and the supposed 
longitudes exact, the probable correction of the parallax indi- 
cated scarcely exceeds 0".l, a quantity too small to be established 
by so small a number of observations. Nevertheless, the example 
proves both that the adopted parallax is very nearly perfect, and 
that a large number of observ-ations at various well determined 
places in the two hemispheres may furnish a good determination 
of the correction which it yet requires. 



Finally, the corrections of the Ephemeris in right ascension 
and declination, according to the above determination of y *^d 
i>, are found by (586) to be (putting a' for a and i' for d) 

A(o — o') = 
A(^ — ^') = 

— 28".42 + 0.469 Ar + 0.187 r^t^et 

— .48 + 0.314 ATT — 0.556 r^tLtt 

This large correction in right ascension agrees with the reenlu 
of the best meridian observations on and near the date of this 
eclipse. Since that time the Ephemerides have been greatly 


To find whether near a given opposition of the moon and Aa 
a lunar eclipse will occur. — The solution of this prob- 
lem is similar to that of Art. 287, except that for 
the sun*s semidiameter there must be substituted the 
apparent semidiameter of the earth*s shadow at the 
distance of the moon ; and also that the apparent 
distance of the centres of the moon and the shadow 
will not be affected by parallax, since when the 
earth's shadow falls upon the moou an eclipse occurs 
for all observers who have the moon above their 
horizon. If S, Fig. 45, is the sun's centre, E that 
of the earth, L3I the semidiameter of the earth's 
shadow at the moon, we have 

Apparent scmidiamotcr of the total 

shadow = LEM 


= BLE—{AES-^ EA V) 

where we employ the same notation as in Art. 287. 

But ob8er\'atiou has shown that the earth's atmosphere 
increases the apparent breadth of the shadow by about its one- 
fiftieth part:* so that we take 

* This fractional increase of the breadth of the shadow was giTen bj Lambibt m 
^, and by Mater as J^. Beer and Madlbr found ^ from a number of obserrmlioM 
of eclip8es of lunar spots in the very favorable eclipse of December 2G, 1833. S«t 
** Der Mond nach teinen koimiachen und individuellen Verhiltnisten, oder ali^rmfmt 
vtrgleichcnde Stlenographie^ von Wilhelm Beer und Dr. Joiuiflf HliXBlCR MJiOLKB,'* 


App. somid. of shadow = — (tt — «' + ^) (^^7) 

In order that a lunar eclipse may happen, we must have, 
therefore, instead of (477), 

P C08 7'< |1 (,r - «'+ ^) + 5 (688) 


or, taking a mean value of i', as in Art. 287, 

/9< [1^ (^ - «'+ ^) +«] X 1.00472 
Employing mean values in the small fractional part, we have 

[S ^"^ - ^+ ^) + ^1 >^ .00472 = 16" 

and the condition becomes 

/9<|^ (^-«'+ ^) + « + 16" (589) 

K in this we substitute the greatest values of ;r, tt', and s, and 
the least value of s'^ the limit 

/9 < 63' 53" 

is the greatest limit of the moon's latitude at the time of opposi- 
tion for which an eclipse can occur. 

K we substitute the least values of ;r, n'^ and s, and the greatest 
value of «', the limit 


is the least limit for which an eclipse can fail to occur. 

Hence, a lunar eclipse is certain if at full moon ^ < 52' 4", 
impossible if ^ > 63' 63", and doubtful between these limits. The 
doubtful cases can be examined by (589), or still more exactly 
by (588), employing the actual values of ;r, ;r', 5, «', at the time, 
and computing /' by (475). ' 

These limits are for the total shadow. For the penumbra we 


App. semid. of penumbra = — (» + ^ + '^) (690) 


80 that the condition (588) may be employed to determine 
whether any portion of the penumbra will pass over the mooiif 
by substituting + 5' for — «'. It will be worth while to make 
this examination only when it has been found that the total 
shadow does not fall upon the moon. 

339. To find the time when a given phase of a lunar eclipse iriB 

occur. — The solution of this problem may be 
derived from the general formulie giren for 
solar eclipses, by interchanging the moon and 
earth and regarding the lunar eclipse as an 
eclipse of the sun seen from the moon ; but the 
following direct investigation is even more 

Let Sy Fig. 46, be the point of the celestial 
sphere which is opposite the sun, or the appar- 
ent geocentric position of the centre of the 

earth's shadow; My the geocentric place of the centre of the 

moon ; P, the north pole. If we put 

a = the right ascension of the moon, 
a = the right ascension of the point S^ 

= R. A. of the sun + 180^, 
d = the declination of the moon, 
d^ = the declination of the sun, 
Q = the angle PSAf, 

L = SM, 

we have 

— d'= the declination of 5, 

and the triangle PS3I gives 

sin X sin Q= cos o sin (a — a') 1 /'«;qn 

sin L cos Q = cos d' sin d -\- sin ^' cos d cos (a — o') J 

The eclipse begins or ends when the arc SM\^ exactly equal to 
the sum of the apparent semidiameters of the moon and the 
shadow. The figure of the shadow will differ a little from a 
cift»le, as the earth is a si)heroi<l; but it will be sufficiently accu- 
rate to regard the earth as a sphere with a mean radius, or that 
for the latitude 45°. This is equivalent to substituting for r iu 
(587) and (590) the parallax reduced to the latitude 45°, which 
may be found by the formula 


TT, = [9.99929] ^ (592) 

where the factor in brackets is given by its logarithm. 

Hence the first and last contacts of the moon with the pe- 
numbra occur when we have 

J^ = ^(.^ + ^+^) + s (593) 

For the first and last contacts with the total shadow, 

Z = ^(n^^s!+^ + s (594) 

For the first and second internal contacts with the penumbra, 

I^ = ^i^. + ^+0-s (595) 

For the first and second internal contacts with the total shadow, 
or the beginning and end of total eclipse, 

i = |l(;r,-5'+^-« (596) 

The solution of our problem consists in finding the time at 
which the equations (591) are satisfied when the proper value of 
L is substituted in them. A very precise computation would, 
however, be superfluous, as the contacts cannot be observed with 
accuracy, on account of the indefinite character of the outline 
both of the penumbra and of the total shadow. It will be suffi- 
cient to write for (591) the following approximate formulse, easily 
deduced from them : 

Zr sin § = (a — a') cos ^ \ 

sin 1" ■ 

Let ns put 

sin 2 ^ sin' 1 (a — a') 

*"" Binr 

a: = (a — a') cos a ) (598) 

a/, y'= the hourly increase of x and y , 
then, if the values of x and y are computed for several successive 

Vol. L— 86 


hours near the time of full moon, we shall also have r? and }f 

from their differences ; and if Xq ^^^ Vq denote the values of x 

and y for an assumed epoch Jij ^^^^ the time of opposition, we 

shall have for the required time of contact T= 2J + r the 


X sin § = rCg + a/r 

jCr cos C = y„ + y'^ 

from which r is obtained by the process already frequently 
employed in the preceding problems. Thus, computing the 
auxiliaries m, JET, n, N^ by the equations 

} (599) 

m%\n M= x^ fiBin N= of 

mQoaM=y^ ncosN=y' 

we shall have 

m sin (M — N) 
sm 4 = ^-i ^ 

L cos 4 m cos (jlf — N) \ (600) 

n n 

in which we take cos '4' with the negative sign for the first contact 
and with the positive sign for the last contact. 

The angle Q = N+ ^^z is very nearly the supplement of the 
angle PMS^ Fig. 46 ; from which we infer that the angle of posi- 
tion of the point of contact reckoned on the moon's limb from the north 
point of the limb towards the east = 180° + N+ '\i/. 

The time of greatest obscuration is found, as in Art. 324, to be 

which is also the middle of the eclipse. 

The least distance of the centres of the shadow and of the 
moon being denoted by J, we have, as in Art. 324, 

A = ±msin{M-^N) (602) 

the sign being taken so that J shall be positive. If then we put 

D = the magnitude of the eclipse, the moon's diameter being 



we evidently have 

D = 

L — A 



in which the value of L for total shadow from (594) is to be 

The small correction e in (598) may usually be omitted, but 
its value may be taken at once from the following table : 

Value of r. 


«— a' 
























































The quantity c has the same sign as ^, and is to be subtracted 
algebraically from d -\- d\ 

Example. — Compute the lunar eclipse of April 19, 1856. The 
Greenwich mean time of full moon is April 19, 21* 5'*.5. We 
therefore compute the co-ordinates z and y for the Greenwich 
times Apr'd 19, 18*, 21*, 24*. 

18» 21* 24* 

5 R.A. 

= a 

0RA. + 


= a' 



a — 

a' (in 


5 Decl. 



= if 

■ c 


log (a — a') 
log cos d 

log X 

13* 46- 36-.62 
13 52 52.98 

— 6 16.36 

— 5645" 

_llo2r 0".2 
+11 35 49 .4 
+ 13 . 

+ 542" 




13*52- 9*.81 
13 53 20.93 

— 1 11.12 

— 1067" 

— 12<> 6'43".7 
+11 38' 22 .8 
— 1701" 




13 53 48.88 
+ 8 56.24 
+ 8544" 

-12^46' 5".5 
+11 40 56 .6 
+ 6. 

— 8903" 




Hence we have the following table : 





— 8** 


= 8/ 


— 5533" 
+ 3456 


x'= + 1498 

+ 542" 

— 1701 

— 8908 

— 2248 

— 2202 


To find L, we have, by (693) and (694), 

rr = 54' 32" r, = 8267" 

«'= 957 
«'= 9 

„j _ s* + ^ 
jV ('f. - «' + ^) 

« = 




L for shadow = 3256 

X for penumbra = 5209 



Assuming the time TJ = 21*, we proceed by (599) and (600) : 

m sin M 

m cos Jtf" 


log m 


210^ 31'.0 

a/ = n sin JV 
if = n cos JV 


+ 1498 
— 741 

116^ 19^.2 



COS (M — JV) 


+ 0*.108 

iTj = Time of middle of eclipse = 21 .108 



log sin -4^ 

L cos 4 


HP 1*.542 

21 .108 


qi 2».881 
21 .108 


19 .566 
22 .650 

18 .227 
23 .989 

For the magnitude of the eclipse, we have, by (602) and (603) : 



m Bin {M — If) = 


= J = 

: 1987" 

L = 



- A — 


2a = 

= 1782 

i> = l?l^ = 0.71 

For the position of the points of contact with the shadow, we 
have, from the above value of log sin 4^ for shadow, taking cos 4^ 
as negative for first and positive for second contact, 

lit Contact. 

2d Contact. 

180»H-JV+ + 

142<» 24' 

116 19 

78 43 

37" 36' 
116 19 
833 55 

and hence 

1st contact is 79® from north point of limb towards the east, 
2d 26® *< " *< " « west. 

The times of the several contacts for any meridian are obtained 
from the times above found by subtracting the west longitude of 
that meridian. 


840. The occultation of a fixed star by the moon may be 
treated as a simple case of a solar eclipse, in which the sun is 
removed to so great a distance that its parallax and semidiameter 
may be put equal to zero. The cone of shadow then becomes 
a cylinder, and the point Z of Art. 289 is nothing more than 
the position of the star, so that the co-ordinates of the moon at 
any time are found by the formulae (482) by regarding a and d 
as the right ascension and declination of the star. In like 
manner the co-ordinates of the place of observation will be found 
by (483). The radius of the shadow is constant and equal to A, 
which is, therefore, to be substituted for L = I — i^ in (490) and 
(491). The co-ordinates z and ^ will not be required unless we 
compute the latter for the purpose of taking into account the 
efiect of refraction according to Art. 327. 

For the convenience of the computer I shall here recapitulate 
the formulse required in the practical applications, making the 
modifications just indicated. 



341. To find the longitude from an observed occidtatum of a star b^ 
the moon. — According to the method of Art. 329, we proceed as 
follows : 

I. Find, approximately, the time of conjunction of the moon 
and star in right ascension, reckoned at the first meridian. Take 
from the Ephemeris, for four consecutive integral hours, two 
preceding and two following the time of conjunction, the moon's 
right ascension (a), declination (d), and horizontal parallax (r). 
Take also from the most reliable source the star's right ascensiou 
(a') and declination (d'). 

For each of these hours compute the co-ordinates x and y by 
the formula 

cos d Sm fa — a ) 

x= ^^ 


_ sin (^ — ^0 cos* i (tt -~ ttp + sin (d + d') sin' j (> — >^ 

sin K 

and, arranging their values in a table, deduce their hourly 
variations x^ and y' for the same instants for which x and y have 
been computed. 

n. Let ji be the local sidereal time of an observed immersion 
or emersion of the star at a place whose latitude is ^, and west 
longitude w ; t the corresponding local mean time. The co-or- 
dinates of the place are to be computed by the formulae 

A sin B = p sin ^' 

A cos B = p cos ^' cos (^fi — o') 

S = P cos f' sin (ai — a') 
7i = A sin(J5 — a') 
C = A co8(J5 — <J') 

When log f is small, add to logs f and rj the correction for 
refraction from the table on p. 517. 

m. Assume any convenient time Tq reckoned at the tir?t 
meridian, so near to t + w that x and y may be considered to 
vary proportionally with the time in the inter\'al / + ai — 7^. 
For the assumed time, take the values of x and y (denoting them 
by To and y©), and also those of x' and y', and compute the aux- 
iliaries m. My &c. by the formulee 


m Bin Jf = ar« — f n sin iV = a/ 


mcosM=y^ — ij ncoQlf=y' 

Bin + = niBiu(M-N) ^ _ ^^3^^^^^ 


where i// is (in general) to be so taken that cos 4^ shall be nega- 
tive for immersion and positive for emersion (but in certain 
exceptional eases of rare occurrence, and of but little use in 
finding the longitude, see Art. 330). Then 

MccoB-^ km cos (M — N) 

n n 

or, when sin i// is not very small, 

hm sin (M — N — 4) 

n sin 4 

K the local mean time / was observed, take h = 3600 in these 
formulse, and then the longitude will be found by 

But if the local sidereal time /i was observed, take h = 3609.856 
in the preceding formulse ; then, /iq being the sidereal time at the 
first meridian corresponding to TJ, 

The longitude thus found will be affected by the errors of the 

IV. To form the equations of condition for correcting the 
longitude for errors of the Ephemeris when the occultation has 
been observed at more than one place, compute the auxiliaries 

T,= T,- ^ {x,Bm N + y.cos N) 
» =: — x^ cos N -\- y^Bin N 


the same value of h being used as before. 

* According to Ovdbmahs {Astron. Naeh., Vol. LI., p. 80), we shoald ase for oocol- 
UtionB k = 0.27264, or log k = 9.485590, which amounts to taking the moon's 
apparent semidiameter about 1''.25 greater in occultations than in solar eclipses. 
Bat it is only for the reduction of isolated obserrations that we need an exact Talue, 
since, when we haye a number of obserrations, the correction of whateyer Talue of 
k we may use will be obtained by the solution of our equations of condition. 


Then, for each observation at each place, compate the coeffi- 
cients V tan '^j V sec 4^, and 

E=yn(t + lit ^ 2\) — «y tan 4 

where o) is the approximate longitude and the unit of < + • — T, 
is one mean hour, and also 

F=ipn^n(t + w-T, ~»vtan4~Avsec4]~ '^^^'^^''^^^"'"^^ 

in which 

i? = f?^ log(l — ec)= 9.99709 

Then, 0' denoting the true longitude, 

m'= m — vy* -J- V tan 4 . * + '^ sec 4 . itLk + ^ . ajt -f- J^ • ^^Aee 
in which f and tf have the signification 

Y = sin iV cos ^ A(a — a') + cos N a(^ — ^') 
* == — cosiV cos d A(a — a') + sin N a(^ — ^0 

The discussion of the equations of condition thus formed may 
then be carried out precisely as in Art. 334, taking y*, 1?, ^ta*, at, 
and ntkce as the unknown quantities. 

Example. — The occultatiou of Aldebaraiij April 15, 1850, was 
observed at Cambridge, Mass., and Konigsberg, as follows:* 

At Cambridge, ^ = 42° 22' 48".6, a; = 4» 44- 30*. 

Immersion, 2* 1* 52*.45 Mean time. 
Emersion, 3 1 38.35 *' " 

At Konigsberg, ^ = 54° 42' 50".4, oi = — 1» 22- 0'.4 

Immersion, 10* 57* 43v66 Sidereal time. 
Emersion, 11 47 47 .60 " " 

I. The Greenwich mean time of conjunction of the moon and 
star was about 7* 30*, and hence we take our data from the 
Nautical Almanac as follows : 

* AstroQoinioal Joarnal, Vol. I., pp. 189 and 175. 



I860 April 16. 





65<> 66' 2rM6 

+ 16° 40' 0".05 

58' 55".22 


66 32 32 .06 

16 46 30 .53 

58 55 .87 


67 8 46 .02 

16 52 54 .77 

58 56 .49 


67 45 3 .02 

16 59 12 .76 

58 57 .10 

The position of Aldebaran for the same date was 


a'= + 16° 12' 1".7 

Hence, by I. of the preceding article, we form the following 







+ 0.58849 

+ 0.47664 

+ 0.10871 


— 0.27671 





+ 0.81176 





+ 0.90014 




n. The sidereal time of Greenwich Mean Noon, April 15, 
1860, was 1* 33"* 8'.96. With this number, converting the 
Konigsberg times into mean times for the sake of uniformity, we 







2* l"»62-.46 

8* 1« 88'.86 

9» 28«» 15'.64 

10* 18« 1K88 

6 46 22.45 

7 46 8 .85 

8 1 15.24 

8 51 10.98 

64<> 2' 2"M 

69* 0' 58".86 

164<> 25' 54". 90 

1760 56' 54".00 

347 12 28.66 

2 11 24.45 

97 86 21 .00 

101 7 20 .10 



















log p sin ^' 

log p cos ^' 


log 7 


The value of log C has been found in order to find the correc- 
tion for refraction. This correction is here quite sensible in the 
case of the Konigsberg observations which were made at a great 



zenith distance. By the table on p. 517, we find that the loga- 
rithms of f and 7] must be increased by 0.000006 for immersion, 
and by 0.000041 for emersion. Appljnng these corrections, the 
values of the co-ordinates are as follows : 



+ 0.02827 

4- 0.57886 

+ 0.54366 


-f 0.44266 

+ 0.48768 

-f 0.80175 

+ 0.88602 

lii. Assuming convenient times not far from t + w^ we have 

Assumed T^ 




8*. 85 


— 0.89440 

+ 0.19406 

+ 0.81176 

+ 0.81188 


+ 0.6686S 

+ 0.67218 

-f 0.69890 

+ 0.78615 

x^ — ^ = m sin if 


+ 0.16579 

— 0.26210 

+ 0.26822 

y^ — tf m cos M 

4- 0.12092 

+ 0.28460 

— 0.10785 

— 0.04987 


297«> 40* 16".6 

850 15' 86". 1 

2470 88' l^.O 

100» 81' 57-.7 






x* — n sin N 

+ 0.68S47 

+ 0.58848 

+ 0.58842 

+ 0.58836 

y* n cosJV 

+ 0.10865 

+ 0. 10857 

+ 0.10856 

+ 0.10S49 


79« 82* 21M 

790 82* 45".8 

790 82' 48^.5 

79*>88' 8-.6 


216 11 85.9 

812 88 59 .0 

167 85 28 .5 21 1 28 .1 

(A — 8600) T 

— 89'. 74 

— 128'.82 

— 68'.68 a-.M 


4* 44"» 37'.81 

4* 44« 12'.88 

— 1*22^ 7'.01 

1 1*22» 4'.90 

iV. For the equations ot condition, taking T^ — 7*.8, 

r, — 7*.2772 It — 3536" 

and putting 

- + .6258 
— the true 1( 


)ngitudo of C 

V — 0.2308 


we find, neglecting terms in A^e, 

w^ = 4» 44- 37'.81 —vr + 1-245 »^ — 2.108 rAA* — 1.293 at 
w. = 4 44 12 .83 — vr — 1.852 * + 2.515 taA' + 1.660 Ar 




;=. — 1 22 7 .01 — vT- — 0.374 .^ — 1.742 rAA- + 0.991 at 
»/= — 1 22 14 .90 — v/- + 0.654 * + 1.822 taA + 1.195 at 

whence the two equations 

= + 24'.98 + 3.097 d^ — 4.623 ttaA — 2.953 at 
= + 7 .89 — 1.028 * — 3.564 ittJi — 0.204 ajt 

If we determine (? and ;rAA: in terms of a^t, these equations give 

* = — 3".33 ^- 0.607 Ar 
^taA = + 3 .17 — 0.232 Air 


and then we find 

w^ = 4» 44- 26-.98 — vr — 0.048 a^ 
<w/= — 1 22 11 .29 — vr + 1169 A:r 

Assuming oi/ = — 1* 22~ 0*.4 as well determined, the last equa- 
tion gives 

vr = — 10'.89 + 1.169 A:r 

which substituted in the value of (o^ gives 

iu^ = 4» 44- 37'.87 — 1.217 att 

Finally, adopting the correction of the parallax for this date as 
given in Mr. Adams*s table (Appendix to the Nautical Almanac 
for 1856), Alt = -\- 5'M, this last value becomes 

«j = 4» 44- 31'.66 

which agrees almost perfectly with the longitude of Cambridge 
found by the chronometric expeditions, which is 4* 44"* 31\95. 
With the same value of aj: we find 

r = — 2".90 * = — (K'.23 Ttiik = + 1".99 

and hence, by (586), the corrections of the Ephemeris on this 
date, according to these observations, are 

A(a —a') = — 2".93 a(^ — a') = — 0".77 


The value niik = + 1''.99 gives aA: == 0.00056, and hence the 
corrected value k = 0.27227 + 0.00056 = 0.27283, which is not 
very different from Oudemans's value. (See p. 551). 

342. When a number of occultations have been observed at a 
place for the determination of its longitude, it will usually be 
found that but few of the same occultations have been observed 
at other places. K, then, we were to depend altogether upon 
corresponding observations at other places, we should lose the 
greater part of our own. In order to employ all our data, we 
may in such case find for each date the corrections of the moon's 
place from meridian observations (see Art. 235), and, employing 
the corrected right ascension and declination in the computation 
of X and y, our equations of condition will involve only terms in 
TT^k and ^tt. The value of A;r will, however, be different on 


diflferent dates, and, therefore, if we wish to retain this term, we 
must introduce in its stead the correction of the mean parallax 
which is the constant of parallax in the lunar tables. K this 
constant is denoted by tt©, we shall have, very nearly, 

Aw = — AW- 

where t: is the parallax for the given date. The equations of 
condition will tihen be of the form 


a = y8ec4 6 = — JP 

In Peirce's Lunar Tables, now employed in the construction of 
our Ephemeris, ;ro= 3422".06. 

343. The passage of the moon through a well determined 
group of stars, such as the PleiadeSj affords a peculiarly favorable 
opportunity for determining the correction of the moon's semi- 
diameter as well as of the moon's relative place, of the relatiTe 
positions of the stars themselves, and also (if observations are 
made at distant but well determined places) of the parallax. 
Prof. Peirce has arranged the formulae of computation, with a 
view to this special application, for the use of the U. S. Coa^t 
Survey. See Proceedings of the American Association for the 
Adv. of Science, 9th meeting, p. 97. 

344. When an isolated observation of either an immersion or 
an emersion is to be computed, with no corresponding obtiena- 
tions at other places, it wnll not be necessarj' to compute the 
values of x and y for a number of hours. It will be sufficient to 
compute them for the time t -\- (o (t being the obsen-ed local 
mean time, and o) the assumed longitude) ; and, as the corre<-tion 
of this time will always be small, the hourly changes may l>e 
found with sufficient precision by the approximate formula?, 
easily deduced from (482), 

xf= — cos ^ V=z 


where da and dd denote the hourly increase of a and d respect- 

345. To predict an occultation of a given star by the moon for a 
given place on the earth. — ^We here suppose that it is already known 
that the star is to be occulted at the given place on a certain 
date, and that we wish to determine approximately the time of 
immersion and emersion in order to be prepared to observe it. 
The limiting parallels of latitude between which the occultation 
can be observed will be determined in the next article. 

For a precise computation we proceed by Art. 322, making 
the modifications indicated in Art. 340. 

But, for a sufficient approximation in preparing for the obser- 
vation, the process may be abridged by assuming that the moon's 
right ascension and declination vary uniformly during the time 
of occultation, and neglecting the small variation of the parallax. 
It is then no longer necessary to compute the co-ordinates x and 
y directly for several different times at the first meridian, but 
only for any one assumed time, and then to deduce their values 
for any other time by means of their uniform changes. It will 
be most simple to find them for the time of true conjunction of 
the moon and star in right ascension, which is readily obtained 
by the aid of the hourly Ephemeris of the moon. Let this time 
be denoted by Tq. We have at this time x = 0, and the value of 
y will be found with sufficient accuracy by the formula 


in which d, ;r, are the moon's declination and horizontal parallax 
at the time Ji, and 8^ is the star's declination. 

Let Aa (in seconds of arc) and a5 here denote the hourly 
changes of the moon's right ascension and declination for the 
time Tq. Then we have, nearly, 


/ A^ 

of — — cos d 

y — — 


^ K 

Let 7\ be any assumed time (which, in a first approximation, 
may be the time Tq itself). Then the values of the co-ordinates 
at this time are 


and to find the time ( T) of contact of the star and the moon*s 
limb, we shall, according to Art. 322, have the following formulae: 

in which /Eij is the sidereal time at the first meridian corresponding 
to Tj, a' is the star's right ascension, and (o is the longitude • 

ABinB = pBin^ ( ^ p cos f'sin ^ 

A cos B = p cos f' cos * Tj = Asin(^B — ^') 

p! = 54148 sin 1" f = /*' A cos J5 

log /i' = 9.41916 1?' = /*' e sin d' 

m sin M=x — f n sinN = af — f 

m cosJtf" = y — ly n cosJV= y — V 

Bm4 = ""'"^^~^ log* = 9.48600 

k cos 4 mcos(3f — N) 

n n 

where '4' i^ to be taken so that cos '^ shall be negative for 
immersion and positive for emersion. 

For a second approximation, we take T as the assumed time 
Tj and repeat the computation for immersion and emersion 
separately. The new value of i> for this second approximation 
will be most readily found by adding the sidereal equivalent of 
T (converted into arc) to its former value. 

It is more especially desirable to know the true time of 
emersion, and the angle of position of the point of reappearance 
of the star. Since this angle in solar eclipses was reckoned on 
the sun's limb, while here it must be reckoned on the moon's, 
it will be equal to 180 + Q: so that, taking the value of i^ froni 
the last approximation, we shall have 

Angle of pi. of contact from the 
north pt. of the moon's limb 

J = 180^ + ivr+ 4 

For the angle from the vertex of the moon's limb, we find ^ by 
the equations 

|) sin /^ = f + f 't pcoBr = Ti + v''^ 


where $, jy, f, rf^ t are to be taken from the last approximation ; 
and then 

Angle of pt of contact fl^m | _ + jy^ + ^ _ 

the Yertex of the moon s limb j ■ i ^ / 

If the computation in any case gives m sin {M— N) > A:, we 

have the impossible value sin -^^ > 1, which shows that the star is 

not occulted at the given place. If we wish to know how far 

the star is from the moon's limb at the time of nearest approach, 

we have (Art. 324) 

J = rh wi sin (M — N) 

the sign being taken so that J shall be positive. This is the 
linear distance of the moon's centre from the line drawn from 
the place of observation to the star, and therefore the angular 
distance as seen from the earth is nJ. The apparent semidiameter 
of the moon is nk^ and hence the apparent distance of the star 
from the moon's limb is ;r(J— A).* 

Example. — ^Find the times of immersion and emersion in the 
occultation of Aldebaran^ April 15, 1850, at Cambridge, Mass. 

The elements of this occultation have been found on p. 558, 
with which an accurate computation may be made by the 
method of Art. 322 ; but, according to the preceding approximate 
method, we proceed as follows. The Greenwich time when the 
moon's right ascension was equal to that of the star is found, 
frt)m the values of a on p. 553, to be 

2; = 7*.47 = 7* 28- 12*. 

For this time we have 

Att = 4- 2174'' ^ = + le** 49' 31".l 

A^ = + 384 d'z= 16 12 1 .7 

^= 3536 ^ — ^'=+ 2249" 

whence, by the above formulee, 

y^ = + 0.6360 a/= + 0.5886 2^= + 0.1086 

Then the computation for Cambridge, tp = 42° 22' 49", 
01 = 4* 44* 30*, will be as follows. For the first approximation, 
we assume 7\ = 7i, and hence we have 

* More exactly, allowing for the augmentation of the moon's semidiameter, it is 
IT (J — *) (1 -J- ^ sin tt), where we have ^ = ^ cos (^ — i'). 




7* 28- 12*. 
Sid. time 6r. noon = 1 33 9 .0 
Eeduction for T^= 1 13 .6 

M,= 9 2 


a'= 4 27 


la =: 4 44 


^ — a'— iii = * = 23 60 46.8 

= 357* 4r.6 

with which we find the following results : 

X = 


f — 

— 0.0298 

m sin Jtf" = 

+ 0.0298 


8" 82'.4 

3> = 

+ 0.6886 

f' = 

+ 0.1940 

n sinJV — 

4- 0.8946 


74» 19'.1 

log sin 4 = 


mcos(Jlf - 


: — 0M690 

For immersion. 


= — 0» 



= 7 



— 6 



— 6» 

49- 15' 


— 4 

44 30 

Local time 

— 2 

4 45 

y = 

+ 0.6360 

7 = 

+ 0.4377 

m cos 3f = 

+ 0.198S 

log m — 



+ 0.1086 


— 0.0022 

ncosJV = 

+ 0.1108 

log 11 = 


log cos 4 — 


A: cos 4 

7 0^.4801 


For emenioB. 

r — + 0*.8111 

^1 = 


T = 

7 .7811 

T — 

7» 46- SS* 


4 44 30 

.ocal time — 

3 2 22 

These times are nearly correct enough ; but, for a more accurate 
time of emersion, we now assume T^= 7*.7811, with which we 

a; = a/ (2; — 2;) = + 0.1831 

^(2;— T;\= +0.0338 
y, = + 0.6360 

y = -f 0.6698 

and to find the new value of t> we have r = + 0*.8111 = 18*40*, 
the sidereal equivalent of which is 18* 43*. 1, or in arc 4° 40'.8. 
This, added to the above value of i?, gives the corrected value 
I? = 2° 22'.4. Repeating the computation with these new values 
of X, y, and i?, we find 


r-oo^(^-J^ = _ 0..5082 * = I"' ^2' 

n N= 74 66 

^^^«* = + .4901 ^^' 

n • 212 17 

T = — 0.0181 r= 3 33 

2\= 7 .7811 208 34 

!r== 7.7630 , ^, ^ xniooi»T/ 
7»4>17« ( ^*^® ^**' reappears at 212® 17' 

T 1 x« o 1 1^ ifrom the north point, or 208® 34' 

Local time = 3 1 17 / « ., . ^ „\, 

jn'om the vertex, of the moons 


This time agrees within 2V with the actually observed time of 
emersion (given on p. 552). The principal part of the difference 
is due to the error of the Ephemeris on this date. 

846. To find the limiting parallels of latitude en the earth for a 
given occuUation. — The limiting curves within which the occulta- 
tion of a given star is visible may be found by the general 
method given for solar eclipses, Art. 311, which, of course, may 
be much abridged in such an application. But, on account of 
the great number of stars which may be occulted, it is not pos- 
sible to make even this abridged computation for all of them. 
The extreme parallels of latitude are, however, found by very 
simple formulae, and may be used for each star. 

For a point on the limiting curve, the least value of J in Art. 
824 is in a solar eclipse = i, but in an occultation it is = k. 
Hence we have, by (557), the condition 

dr m sin {M — i\r) = k 

or, restoring the values of m sin M= x — $, m cos J!f = y — tj^ 

(x — ^) cos N — (y — )y) sin iV = it ^ 

The angle iVis here determined by the equations (552); but, for 

an approximate determination of the limits quite sufficient for 

our present purpose, we may neglect the changes of f and jy, and 


nsiniV=a/ ncosiV = y' 

Let Xq and y^ be the values of x and y for the assumed epoch 
Tq ; then for any time T= TJ + r we have 

X == rTg 4" ^ sin iV . r y = y^ -|- n cos JSf . t 

Vol. L— 36 

— ■ •-■& .• — >, — T sin -V = — k 

- -i i liz-.a :i Tv* - 'w'-r Liivc. by negiectin^ the coin- 
er: j- = -r :-:s 3' — Z sin -5' 

.^ - _-~ T r V ttrcmiize the maximum and mininom values 

^ ._ L^ Ilr^ JCC'iii-jELS. Let Us put 

f = — ^ cos y ^ r^ sin X 
> = c *in 3' -r 15 cos -Y 

c = — a cos -\' -p 6 sin JV 
7 = a sin J\' -|- ^ cos -V 
: = ^ . 1 _ a' — 6«; 

"•Six «« *»<^ oa^^- bv our first condition, 

tz = — J, COS -y + y, sin -Y ±: it 
^ , vi!>ni:i: ^-lanritT, since wc may here assume jr' and / 

>^ ^» *\ Jiav: r — :-— ^= 1, we can assume y and e so as to 

COS / = a 
sin ;* cos t = b 
sin 7 sin c = C 

J »:'.ii«i si-i r •-'^^^' ^'^^ restricted to positive vahies. The formula 

^ , c^ r *iti -V cos J' + sin 7^ cos e cos Xcos d' + sin r sin e sin S' 

^^.:t »d^j" K* put under a more simple form by assuming fi and 
j^ « CO ^dkCislY the conditions 

sin t^ = sin iV cos $' 
cos ,3 cos X = cos N cos d' 
cos j3 sin A == sin d' 

^^kieA ^x^ i^ i^^^^y ^^ restricted to positive values. 


We thus obtain 

sin f = sin /9 cos y -f cos /9 sin y COS {X — e) 

in which ip and e are the only variables. Since cos j9 sin y is 

positive, this value of sin ^ is a maximum when cos (X — e) = 1 

or X — e = ; and a minimum when cos {X — e) = — 1, or 

X — e = 180°. Hence we have, for the limits, sin ^ = 8in(^ ± y)^ 

that is 

for the northern limit, f> = ^ -(- /* 

for the southern limit, 9 = P — y 

One of the points thus determined may, however, be upon 
that side of the earth which is farthest from the moon, since we 
have not restricted the sign of ^, and our general equations 
express the condition that the point of observation lies in a line 
drawn from the star tangent to the moon's limb, which line 
intersects the surface of the earth in two points, for one of which 
(^ is positive and for the other negative. But, taking ^ only with 
the positive sign, we must also have sin e positive. For the 
northern limit, therefore, when A = e, sin ^ must be positive, 
which, according to the equation cos ^ sin A = sin 5', can be the 
case only when 5' is positive. Hence the formula <p = fi + T 
gives the most northern limit of visibility only when the star is 
in north declination. For similar reasons, the formula <p = ^ — y 
gives the southern limit only when the star is in south declina- 
tion. The second limit of visibility in each case must evidently 
be one of the points in which the general northern or southern 
limiting curve meets the rising and setting limits, — namely, the 
points where ^ = 0, and consequently, also, sin e = 0, cos e = ± 1, 
which conditions reduce the general formula for sin ^ to the 
following : 

Bin f) = (sin N cos y dz cos iV sin y) cos ^' = sin (N ±: y) cos d' 

K cos iVis taken with the positive sign only, the upper sign in 
this equation will give the most northern limit to be used when 
the southern limit has been found by the formula <p = ^ — y, and 
the lower sign will give the southern limit to be used when the 
northern limit has been found by the formula <p = fi -\- y. 

Finally, since the epoch T^ is arbitrary, we may assume for it 
the time of true conjunction in right ascension when 0^)= 0, and 
we shall then have 

a = cos /' = y^ sin iV ± A' 


The above discussion leads to the following simple arrangement 
of the formulfiB 

COS r^ = y„ sin N ±i 0.2723 (r < ISO**) 

sin ^ = sin i\r cos d' (/9 < 90*) 

9, = fi±ri ) (604) 

cos rt = j/o sin iV =F 0.2723 
Bin ^, = sin (iV rp r») cos ^' (iV< 90<*) 

in which the upper or the lower signs are to be used, according 
as the declination of the star is north or south. When the 
declination is north, ^^ will be the northern limit and y, the 
southern ; and the reverse when the declination is south. The 
angle Nis here supposed to be less than 90°, and is found by 
the formula 

taniV = ^ 

always considering y' as well as x' to be positive. 

When the cylindrical shadow extends beyond the earth, north 
or south, we shall obtain imaginary values for y^ or y^ The 
following obvious precepts must then be observed : 

1st. When cos 7*1 is imaginary, the occultation is visible beyond 
the pole which is elevated above the principal plane of reference, 
and, therefore, we must put for the extreme limit <fi^^-r 90®, or 
<fi= — 90°, according to the sign of 5'. 

2d. When cos y^ is imaginarj^ the value of ^^ will be the lati- 
tude of that point of the (great circle) intersection of the prin- 
cipal plane and the earth's surface which lies nearest the depressed 
pole; that is, we must take ^^=d'—90^, or f, — o'-!-90% 
according as o' is positive or negative. 

It is also to be observed that the numerical value of f| 
obtained by the formula <p^^= ^ ± y^ may exceed 90°, in which 
case the true value is either f^= 180° — (^ ±: ^,), or yj= — 180^ 
— (/3 =i= yi)y since these values have the same sine. 

Example. — Find the limiting parallels of latitude for the 
occultation of Aldebaran^ April 15, 1850. 

We have found, page 559, for this occultation, 

y^= + 0.6360 x' = 0.5886 ^ = 0.1086 

Hence, with 3' == 16° 12', we find 



N = 79^ 33' log sin /9 = 9.9751 

y, sin iV = + 0.6255 p = 70^ 47 

k= 0.2723 ri = 26 8 

cos r, = + 0.8978 /9 + ^'j = 96 55 

cos z', = + 0.3532 S^i = 83 5 
r, = 69<' 19' 

N—U= 10 14 ?>,= 9 49 

It is hardly necessary to observe that the occultation is not 
visible at all the places included between the extreme latitudes 
thus found, since the true limiting curves do not coincide with 
the parallels of latitude, but cut the meridians at various angles, 
as is illustrated by the southern limit in our diagram of a solar 
eclipse, p. 504. Unless a place is considerably within the 
assigned limits, it may, therefore, be necessary in many cases to 
make a special computation, by the method of Art. 345, to deter- 
mine whether the occultation can be observed. 


847. If the disc of a planet were always a circle, and fully 
illuminated, its occultation by the moon might be computed by 
the general method used for solar eclipses by merely substituting 
the parallax and semidiameter of the planet for those of the sun ; 
and this is the method which has generally been prescribed by 
writers on this subject. But with the telescopes now in use, 
and especially with the aid of the electro-chronograph, it is 
possible to observe the instants of contact with the planet's limb 
to such a degree of accuracy that it appears to be worth while 
to take into account the true figure of the visible illuminated 
portion of the planet. Moreover, the investigation of this true 
figure possesses an intrinsic interest which justifies entering upon 
it here somewhat at length. 

In order to embrace at once all cases, I shall consider the 
planet as a spheroidal body which even when fully illumi- 
nated presents an elliptical outline, and when partially illumi- 
nated presents an outline composed of two ellipses, of which 
one is the boundary of the spheroid and the other is the limit of 
illumination on the side of the planet towards the observer. 1 
begin with the determination of the first of these ellipses. 


348. To find the apparent form of (lie disc of a spheroidal planet* — 
Let us first express the apparent place of any point of the 
surface of the planet, by referring it to three planes perpen- 
dicular to each other, of which the plane of xy coincides with the 
plane of the planet's equator, while the axis of z coincides with 
the axis of rotation. In this system, let 

x,y,z = the co-ordinates of the point on the surface of the 

f , iy, C = those of the observer. 

Straight lines drawn from the observer to the centre of the 
planet and to the point on its surface determine their apparent 
places on the celestial sphere. If these places are referred to 
the great circle which corresponds to the planet's equator, and 
if we put 

X, X' = the geocentric longitudes of the apparent places of the 
planet's centre and the point on its surface, reckoned 
from the axis of x, in the great circle of the planet's 

/5,/9'=the latitudes of these places referred to the great 
circle of the planet's equator, 

p^ /o' = the distances of the centre of the planet and the point 
on its surface from the observer, 

we shall have (Arts. 32 and 33)t 

p cos fi cos X == — $ 

p cos /9 sin >l = — rj y (605) 

/o sin ^ = — C 

p' cos fi' cos X' = X — $ 

p' cos /?' sin i' = y — 13 )- (606) 

/o' sin ^ = 2 — C 

* The method of investigation here adopted, so far as relates to the apparent form of 
the disc, is chiefly derived from Bessel, Astronomische Unterauchungen^ Vol. I. Art. VI. 

f The group (606) may be deduced by supposing for a moment that the position 
of the observer is referred to a system of planes parallel to the first, but having its 
origin at the point on the surface of the planet. The co-ordinates in this system are 
equal to those in the first increased respectively by x, y, and z. The negative sign 
in the second members of both groups results from the consideration that the longi- 
tude of the observer as seen from the planet is ISO^-}- ^ ^^ 1S0^4~ ^'i ^^'^ ^^ 
latitude, — ^, or — ^'. Compare Art. 98. 

VORM OF A planet's DISC. 


Now, let and C, Fig. 47, be the apparent Fig. 47. 

places of the planet's centre and the point on its 
surface, projected upon the celestial sphere; Q 
the pole of the planet's equator; Pthe pole of the 
earth's equator ; and let 

^ = the apparent distance of C from = the arc 

|/= the position angle of G reckoned at 0, from 

the declination circle OP towards the east, 

= POCy 

p= the position angle of the pole of the planet 
= POQ; 

then, in the triangle §0(7, we have 

sin ^ sin (j/ — P) = cos /9' sin (X' — X) 

sin s^ cos (p' — p) = cos fi sin /9' — sin /9 cos /9' cos (A' — X) 

Multiplying these by /t>', and substituting the expressions (605) 
and (606), we obtain 

p' sin s^ sin (p' — p) = 
p' sin «* cos (y — p) = 

— X BinX + y cos X 

— X sin /9 cos X — y sin /9 sin A -f ^ cos fi 

or, since s' is very small and /t>' sin 5' or /o's' differs insensibly 
jfrom josin 5' or ps\ 

ps' sin (p'—p) = — a: sin A 4- y cos A ) 

ps^ cos (y — 1>) = — X sin ^ cos X — y sin ^ sin il -|- ^ cos fi ) ^ ^ 

These equations apply to any point on the surface of the planet. 
K we apply them to those points in which the visual line of the 
observer is tangent to that surface, they will determine the curve 
which forms the apparent disc. The equation of an ellipsoid of 
revolution whose axes are a and 6, of which b is the axis of 
revolution, is 


1= — 4--^+- 

and the equation of a tangent line passing through the point 
whose co-ordinates are $, ij, and ^ is 

1=^.1- y? + !? 

aa aa hb 


The distances f, rj, and C are very great in comparison with x, 


y, and z. K we divide (609) by />, the quotients -? ^ - will be 

X v z X 

of the same order as -, -, ^, but the quotient - will be inappre- 

X V z 

ciable in relation to the quotients — ? -=^, -rr-- Performinfir this 

^ aa aa ho ° 

division, therefore, and substituting the values off, gy, and Z from 

(605), we may write for the equation of the tangent line 

f. X cos p cos k , y cos a%uik , z sin 3 ,/»,^x 

= h ^ ^ j^ (610) 

J£ the curve ACBy Fig. 47, is referred to rectangular axes 
passing through the apparent centre of the planet, one of 
which is in the direction of the pole of the planet, and if u and 
V denote the co-ordinates of any point of the curve, so that 

u = 5' sin (/?' — p) 
V = ^ cos (y — p) 

the equations (607) and (610) will enable us to determine x, y, 
and z in terms of u and v. Putting 

bb , 

— = 1 — ee 


the three equations become 

pu = — X Bin X -\- y cos X 
pv = — (x cos ^ + y sin X) sm fi -{- z cos fi 
= (xcoB X -{- y sin X) (1 — ee) cob fi + z ^n fi 

from which we derive 

— X sin X -{- y COB X = pu 

, . , sin /9 

— X cos X — y sin X = pv 

1 — ee cos'^ 

fl — ee) cos 8 

z = pv ^ ^ 

1 — ee cos'/9 

Substituting these values in (608) and putting 

s = - = the greatest apparent semidiameter of the planet, 

c = T|/(l — ee cos'/5) 

FORM OF A planet's DISC. 569 

we find 

55 = titt + - (611) 

which is the equation of the outline of the planet as projected 
upon the celestial sphere, or upon a plane passed through the 
centre of the planet at right angles to the line of vision. It 
represents an ellipse whose axes are 25 and 25 |/(1 -— ee cos^/S), 
e being the eccentricity of the planet's meridians. The minor 
axis {OBy Fig. 47) lies in the direction of the great circle drawn 
to the pole of the planet's equator. 

We next proceed to determine what portion of this ellipse is 
illuminated and visible from the earth. 

849. To find the apparent curve of illummation of a planefs surface. — 
If the sun be regarded as a point (which will produce no sensible 
error in this problem), the curve of illumination of the planet, as 
seen from the sun^ can be determined by conditions quite similar 
to those employed in the preceding problem ; for we have only 
to substitute the co-ordinates expressing the sun's position with 
reference to the planet, instead of those of the observer. If, 
therefore, we put 

A, B = the heliocentric longitude and latitude of the centre 
of the planet referred to the great circle of the 
planet's equator, 

the equation of the tangent line from the sun to the planet, 
being of the same form as (610), will be 

^ xcobBcoqA , vcosJ^sinyl . zainB 

= h ^ j^ — (612) 

aa * aa ^ bb ^ ^ 

If each point which satisfies this condition be projected upon 
the celestial sphere by a line from the observer on the earth, and 
u and V again denote the co-ordinates of the projected curve, we 
have here, also, to satisfy the equations 

pu = — X sm X -}- y cos X | 

pv = — (xco&X -{- y sin X) sin iS + z cos fi j (y V 

in which X and ^ have the same signification as in the preceding 
article. The values of x, y, and Zy determined by the three 
equations (612), (613), being substituted in the equation of the 
ellipsoid, we obtain the relation between u and v, or the equation 


of the required curve of illumination as seen from the earth. In 
order to fecilitate the substitution, let us put 

a:, == — a: sin A -f y cos X 
y^ = X cos A 4" y sin i 

from which follow 

x= — Xi sin ^ + yi cos X 

yz= Xi cos A + y, sin X 

At the same time, let us introduce the auxiliaries fi^ and B^ 
dependent upon ^ and B by the assumed relations 

- cos A = cos B — cos j^j = cos B 
g Cr 

- sm a,= r- Bin fi -^ sm Bi= t muB 
go Gr 


Then the three equations become 


= X, cos By sin (-4 — A) -(- y^ cos B^ cos {A — ^) + I ^ ^^^ ^i 

fiU = Xi 

-gpv = ^ yi sin fii + i^ cos fi^ 
from which we derive 

Xi= pU 

Ny^ = — pu cos /5j cos Bi sin {A — X) — - gpvBinBx 

Nt z = — /ou sin /5iC08 ^,8in (^1 — ^)-{-Tgp^ cos J5iC0S(yl — X) 

where, for brevity, iVis put for sin ^^ sin B^ + cos^j cos B^ cos {A — >l). 
Before substituting these expressions in the equation of the 
ellipsoid, it will be well to consider the geometrical signification 
of the quantities /9i and By If we draw straight lines from the 
centre of the planet to the earth and to the sun, the latitudes of 
the points in which these lines intersect the surface of the planet 
will be /? and B. If these points be projected upon the surface 
of a sphere circumscribed about the ellipsoid, by perpendiculars 
to its equator, the latitudes of the projected points will be ^, and 
jBj ; and g and G will be the corresponding radii of the ellipsoid. 
If now these projected points are referred to the celestial sphere, 
by lines from the planet's centre, they will form with the pole Q 
of the planet's equator a spherical triangle QOSy in which the 

rORM OF A planet's DISC. 571 

angle Q will he A — X; and the sides including this angle will 
be 90° — A = QOy 90° — B,= QS. Denoting the angle at by 
1(7, and the side OS by V, we shall have 

cos F = sin ^1 sin Bi + cos fi^ cos Bi cos (A — X)^ 
sin V cos w = cos A sin J?, — sin fi^ cos J?^ cos (A — X) > (616) 
sin F sin w = cos j?^ sin (yl — ^) J 

in which V is very nearly the angular distance between the sun 
and the earth as seen from the planet. 
This triangle also gives 

sin J5j = cos V sin fi^ -(- sin V cos fi^ cos w 
cos B^ cos (A — X) = cos F cos ^j — sin F sin fi^ cos w? 
cos B^ sin (il — 'I) = sin F sin lo 

By these equations the above expressions for x^, y^, and ^ are 
reduced to 

cos F. x^ = /DM cos F 

cos V.yi = — pu sin F sin ti? cos ^^ 

— r S'/oy (cos F sin /9j + sin F cos ^^ cos u?) 
cos V'j'Z = — pu sin F sin w sin /9j 

Substituting these in (608), observing that zx + yy = x^x^^ + y^^/j, 
we have 

cos'F— = uu cos'F 

+ I (u sin M? + T S'^ cos w) sin F cos /?i + r ^^ ^ob V sin ^^ I 

-|- I (m sin i^ + - l/y cos w) sin F sin fi^ — j- gv cos F cos ^j I 

Developing the squares in the second member, and putting s for 

-, and also 

€ = i/Cl — ee cos* /5) = — 

we shall find 


= I M cos w? — V I + I M sm M? + r j sec* F (616) 



which is the required equation of the curve of illumination, as 
seen from the earth, projected upon the celestial sphere. It 
represents an ellipse whose centre is at the origin but whose 
axes are, in general, inclined to the axes of co-ordinates, and, 
consequently, to the axes of the ellipse of equation (611). The 
equation (611) is only the particular case of (616) which corre- 
sponds to V= 0, or the case of full illumination. 

Fig. 48. 

350. We have yet to determine what portions of the apparent 

disc are bounded by the two curves 
respectively. K ABA'B^, Fig. 48, 
is the ellipse of (611), which I shall 
call the first ellipse, and CDC'D' that 
of (616), which I shall distinguish as 
the second ellipse, the visible outline 
of the planet is composed of one- 
half the first and one-half the second 
curve, and these halves either begin or end at the points C and 
C, which are the common points of tangency of the two curves. 
These points satisfy both equations ; and, therefore, putting Wj and 
1?! for the co-ordinates of either point, and subtracting (611) from 
(616), we find 

A / • . cos tr W , T^ 
= I M, sm t^ + Uj j tan' V 

which is satisfied, in general, by taking 

cos w ^ 
w, sm u? + r, = 

Denoting the position angle corresponding to u^ i\, by p^, we 
have 2/i= 5i 8in(pi — 7?), Vi=^ Sy co^^pi—p). Substituting these 
values, and also putting 

c, sm ic, = sin w 

c^ cos tr, = 

cos w 


the preceding condition becomes 


Cj s^ cos (^j — p — tTj) = 
p^ = p + ir, :;: 90« 


which expresses the position angles of both Cand C K we 
draw the arc 0D0% Fig. 48, making the angle BO(y = w?, and 

FORM OP A planet's DISC. 673 

take 00' = Vj the point 0' will be nearly the position of the 
planet as seen from the sun, and the arc V will be the measure 
of the angular distance between the sun and the earth as viewed 
from the planet. If we assume sin w to be positive in equations 
(615), as we are at liberty to do, the arc 'Fwill be reckoned from 
the planet eastward from 0° to 860°. Now, so long asFis less 
than 180°, the west limb will evidently be the full limb, and 
when y is greater than 180°, the east limb will be the full limb. 
Hence we infer that a point whose given position angle is p' is 
on the east limb when 

P'>P + ^1 — 90° and < i? + w^ + 90° 
but on the west limb when 

P' < i? + w?i — 90° and > p -f t/?^ + 90° 

When V > 90° and < 270°, the planet is crescent ; but when 
'F> 270° and < 90°, it is gibbous. In the case of a crescent 
planet there are two points, one on the full and the other on the 
crescent limb, corresponding to the same position angle : hence 
in observations of a crescent planet the point of observation on 
the limb will not be sufficiently determined by the position 
angle alone ; it will be necessary for the observer to distinguish 
the crescent from the full limb in his record. 

351. In order to apply the preceding theory, it is necessary to 
find the quantities /), /9, >l, -B, A, Tlie direction of the axis of x 
in Art. 348 was left indeterminate, and may be assumed at 
pleasure, but it is most convenient to let it pass through the 
ascending node of the planet's equator on the equinoctial, so that 
X and // will be reckoned from this node. The position of the 
node must, therefore, be kno^vn, and this we derive from the 
researches of physical astronomers. If we put 

n = the longitude of the ascending node of the planet's 
equator on the equinoctial, 

i = the inclination of the planet's equator to the equi- 

we have at any given time ^, for the planets Jupiter and Saturn, 
the only ones whose figures are sensibly spheroidal. 


^ (n = 367° 56' 25" + 3".59 (t — 1850) 

J^or jupiter. | . ^ ^50 25' 49" + 0".66(f - 1850) 

„ ^ f n = 125° 13' 54" + 128".76 (/ — 1850) + 0".0605 (t — 1850)« 

j^or&aiurn.| ,^ yoio'lO"— 15".08(f— 1850) + 0".0035(^— 1850)» 

in which t is expressed in years.* 

The values for Saturn apply either to its equator or the rings, 
which are sensibly in the same plane. 

If now we put 

o', d'z=z the right ascension and declination of the planet, 

we can convert a' and 8' into X and fi by Art. 23 ; we shall 
merely have to substitute in (29) or (31) a' — n for a, d' for 8, 
and i for e. The angle p is here the position angle of the pole 
of the planet reckoned from the declination circle of the planet 
towards the east; but in Art. 25 the angle tj is tlie position 
angle reckoned towards the west, and, therefore, we shall have 
to put 7j = 360° — p in (33). Hence we obtain the following 
formula for ^, iy and p : 

fBinF= tan ^' /' sin X = cos (F — i) 

f cosF = sin (a' — n) /' cos k = cos F cot (a' — n) 

tan ^ = sin A tan (jP — i) K (619) 

sin F' cot (o' — n) 

tan F'= tan i sin (o' — n) tan p = — 


To find A and By we avail ourselves of the heliocentric longi- 
tude and latitude of the planets given in the British Almanac, 
and as these quantities are referred to the ecliptic, while A and 
B are referred to the planet's equator, we must know the rela- 
tive position of these circles. Putting 

N*^ the longitude of the node of the planet's equator on 

the ecliptic, 
7'= the inclination of the planet's equator to the ecliptic^ 
N = the arc of the planet's equator between the equi- 
noctial and the ecliptic, 

* These values I have deduced from the data given in Damoi8EAU*8 Tablet icUp- 
tiquea dea Satellites de Jupiter^ Paris, 18^6 ; and Bessel's Bentimmung der Lagt und 
Grbtae dea Satuma-Ringea und der Figur und Groaae dea Satuma^ Aatronom, Xach.^ Vol. 
XII. p. 167. 

FORM OP A planet's DISC. 575 

we deduce fix)m the data of Bessel and Damoiseau, for a given 
year <, 

r N'=SSb^ 40' 46"+ 49".80 (t — 1850) 
PorJupiterJ /'= 2<» 8'51"+ 0".43(^— 1850) 

{n = 336^ 33' 1 8"+ 46".55 (t — 1850) 

{N'= 167^ 31' 52"+ 46".62 (^ — 1850) 
/'= 28^ 10' 27"— 0".35(f— 1850) 
N = 43^ 31' 34"— 86".75 (t — 1850) — 0".0625(< — 1850)« 

and these values for Saturn also apply to the rings. 
Finally, if we put 

A', B'=the heliocentric longitude and latitude of the 
planet, referred to the ecliptic, 

the formulse (29) or (31) will serve to convert A' — iV' and 5' 
into A — N and B ; and they become 

X sin M= tan B' K' sin {A^N) = cos {M — J') 

JT cos Jf = sin (J'— iV'') JC'cosCyl— iVr)==cos Jf cot(il'— iV^') ( (620) 

tan jB = sin (il — N^ tan {M — 7') 

352. The preceding complete theory admits of several abridg- 
ments in its application to the different planets, varying according 
to the features peculiar to each. 

Jupiter. — The inclination of Jupiter's equator to the ecliptic is 
80 small that the quantity c = i/(l — ee cos* ^) never differs 
sensibly from |/(1 — ee), which, according to Struve's measures, 
is 0.92723. I shall, therefore, use as a constant the value 
log c = 9.9672. Again, on account of the small inclinations both 
of Jupiter's equator and of his orbit to the ecliptic, the angle w 
never differs much from 90°, and, since this angle is required 
only in computing the gibbosity of the planet (which never 
exceeds 0".5), it is plain that we may take w = 90°, and that V 
rasLj be found with sufficient accuracy by the formula 

or, indeed, by the formula 

F=il' — ;' (621) 

in which A' and A' are, respectively, the heliocentric and geo- 
centric longitudes of the planet, the former being taken directly 


from the British Almanac, and the latter computed from the geo- 
centric right ascension and declination by Art. 23 : so that for 
this planet the equations (615), (619), (620) will be dispensed 
with, except only the last two equations of (619), which will be 
required in finding 'p. 

Saturn. — The inclination of Saturn's equator to the ecliptic is 
over 28°, and therefore the quantity c = v (1 — ^^ cos*^) will 
have sensibly different values at different times. The value of 
— ^ is, however, given in the table for Saturn's Ring in our 
Ephemerides (where it is usually denoted by l). The value of et 
is 0.1865, or log ee = 9.2706. The gibbosity of Saturn is alto- 
gether insensible ; so that we shall have occasion to use only the 
equation (611), or in any formula that may be derived from the 
more general equation (616) we shall have to putF= 0. The 
angle p is also given in the table for the ring. 

Saturn's Ring. — The ring may be here regarded as an ellipsoid 
of revolution whose minor axis = 0. Hence we have only to 
make e = 1 in our formulae to obtain the equation of its elliptical 
outline. This gives c = i/(l — cos*^) = sin ^, which value being 
substituted in (611), we have at once the required equation, 
while the position of the ellipse is given at once by the angle p 
from the table above referred to. 

Mars, Veniis, and Mercury. — These planets may be regarded as 
spherical in the computation of their occultations, and we shall, 
therefore, have to consider only their crescent and gibbous 
phases. To adapt our formulae to the case of a spherical body, 
we have only to put c = 0, or c = 1. Since in this case we are 
concerned only with the apjmrcni figure of a partially illuminated 
spherical body, we may, for the convenience of computation, 
assume any point as the pole of the planet ; and it will be most 
natural to assume the point which is the pole of the great circle 
whose plane passes through the sun, the earth, and the planet 
p. ^g The direction of this pole is evidently the 

same as that of the line joining the cusps 
of the partially illuminated disc. This makes 
^ == 0, ^ = 0, in (615), and, consequently, 
F= A — X. But, as the adopted equator of 
the planet is here a variable plane, we can 
no longer use the form (620) for finding A. 
A very simple and direct process for finding 
V offers itself. Let E, S, 0, Fig. 49, repre- 

PORM OP A planet's DISC. 577 

sent the centres of the earth, the sun, and the planet; S'0'0", 
the great circle of the celestial sphere whose plane passes through 
the three bodies ; S' and (y, the geocentric places of the sun and 
the planet ; 0", the heliocentric place of the planet. Then Cy 0" 
is the arc heretofore denoted by V, and, in the infinite sphere, is 
the measure of the angle OOO"' = SOK Putting then V=0'0", 
Y = S'Cy^ and also 

R' = SO = the heliocentric distance of the planet, 
R =8E= « " « earth, 

we have 

Sin F = — sin y 

We might find V directly from the three known sides of the 
triangle SOE; but, as we have yet to find p, and y comes out at 
the same time with |) in a very simple manner, it will be prefer- 
able to employ the above form. 

To find p and 7-, let S', 0", 0", Fig. 50, be the three places 
above referred to, and P the pole of 
the equinoctial. Draw CyQ perpen- 
dicular to the great circle S^(yO''. 
This perpendicular passes through the 
adopted pole of the planet, and we 
have P<yQ=p, or PO'S'= 90° — p, 
and S'(y=y. Hence, denoting by 5' g, 
and D the declination of the planet 
and the sun, and by a' and A their 
right ascensions respectively, the spherical triangle PS'O' gives 

Y = sin d* sin D + cos 5' cos D cos (»' — A) \ 

p = cos d' sin D — sin d' cos D cos (a' — A) \ (622) 

cos y = 
sin y sin p 
sin y cos p = cos D sin (a' — A) } 

Hence, introducing an auxiliary to facilitate the computation, 
both p and V will be found by the following formulse : 

tan F = tan D sec (a' — A) 

tan p = cot (o' — A) sin (F — d') sec JP 

. -- R sin (a' — ^) cos D ( (^^^) 

sm V = -— ^^ ^ 

R' C08p 

In this method of finding V we do not determine whether it is 

Vol. L— 37 



greater or less than 90°. This is of no importance in computing 
an actual observation, but only in predicting the phase of the 
planet, whether crescent or gibbous. For the latter purpose we 
must have recourse to the triangle SUO of Fig. 49, the three 
sides of which are given in the Ephemerb. 

The value of V being found, the equation (616) will be used to 
determine the apparent outline after substituting c = 1 and 
w = 90°, whereby it becomes 

The value of s in our equations is supposed to be given. It 
will be most convenient to deduce it from the apparent semi- 
diameter of the planet when at a distance from the earth equal 
to the earth's mean distance from the sun, which is the unit 
employed in expressing their geocentric distances in the Ephe- 
meris. Thus, denoting the mean semidiameter by s^ and the 
geocentric distance by r', we have (Art. 128) 



and Sq may be taken from the following table : 






Saturn's Rings 

Outer semi-major axis of outer ring 

Inner «* " ** «' 

Outer ** " inner " 

Inner " *« " «« 





















Lb Verrier, Theory ofMerewnf. 
Peircs, Am. Ephemeru. 




Struvb, A$tT. Naeh., No. 139. 
Bessel, Attr. Nach., No. 276. 




f Struve, Attr, A'acA.jNo. 139, 
reduced to agree with Bes- 
8EL*s measures of the outer 
diameter of the outer ring. 

353. To Jiiid the longitude of a i^lace from the obsei*ved contact of 
the moon's limb with the limb of a planet. — In the following investi- 
gation, it is assumed that the quantities p, m?, V^ c, are known for 
the time of the oceultation. They may be computed by the 
above methods for the time of conjunction of the moon and 
planet, and regarded as constant for the same oceultation over 
the e^rth in general. 



Let 0, Fig. 51, be the apparent centre of the planet, and G 
the point of contact of its limb with 
that of the moon. Let OJ!f be drawn 
from towards the moon's centre, in- 
tersecting the moon's limb in D. Since 
the apparent seraidiameter of any of 
the planets is never greater than 31", 
it is evident that no appreciable error 
can result from our assuming that the 
small portion CD of the moon's limb 
coincides sensibly with the common 
tangent to the two bodies drawn at C 
K, then, the planet were a spherical 
body with the radius OD, the observed 

time of contact would not be changed. We may, therefore, 
reduce the occultation of a planet to the general case of eclipse 
of one spherical body by another, by substitutmg the perpen- 
dicular OD for the radius of the disc of the eclipsed body. Let 
s" denote this perpendicular; let OA and OQ be the axes of u 
and V respectively, to which the curve of illumination is referred 
by the equation (616) ; and let i? be the angle QOD which the 
perpendicular s" makes with the axis of r. The equation of the 
tangent line CD referred to these axes is 

u sin ^ -}- t; cos * = «" 

"We have also in the curve 


T- = — tan iJ 


Differentiating the equation (616), therefore, we have 

(v sin M7 \ / . tan y% sin w \ 
u cos w 1 1 cos w -\ I 


, / . . t?co8M?\/ . tan»9co8tr\ ._- ^ 
-f I ti sm w -\ M ®*^ ^ I ®®^ F = 

By means of this equation, together with (616) and (625), we can 
eliminate u and i?, and thus obtain the relation between 8 and 5". 
To abbreviate, put 

V sin w 

X = u cos w — 

y = u sin u? -[- 




and also 

c' Bin d' = — c' cos i>' = cos * (626) 

then the three equations become 

X cos (d' — w) — y sin (v>' — w) sec«F = 


a: sin (i^' — tr) + y cos (v>' — u?) := — ^ 

From the first and second of these we find 

s sin (y — w) 

X = 

y = 

|/ [1 — co8» (i*' — \D) sin« F] 

5 COS (*' — XD) COS* F 

l/[l — cos»(d'— M7) 8in»F] 

which substituted in the third give 

5" = 5CC V[l — cos' (v>' — M?) sin'F] 

Hence, if we put 

sin / == cos (»>' — u?) sin F 
we have \ (627) 

5" = 5.CC'C08/ 


TVo have seen (Art. 352) that in all practical cases we may take 
w = 90°, and, therefore, instead of (626) and (627) we may 

employ the following : 

„, tan »J 

tan »^' = 


sin / = sin t>' sin F ) (628) 

s sin f^ cos / 


sin y 

If the occultation of a cusp of Venus or Mercury is observed, 
we have at once 5"-- s cos «? (for the axis of v coincides with 
the line joining the cusps), and we do not require F. 

The value of s" is to be substituted in (486) for the apparent 
semidiameter of the eclipsed body. In that formula, ^denotes 
the apparent semidiameter at the distance unity : therefore, we 
must now substitute the value 

sin jy == r' sin «" 


or, by (624) and (628), 

. „ sin 5. sin * cos y .^^. 

sin JU = ®— r — (629) 

sin d' ^ ^ 

Since / is here very small, we may put tan / = sin /, and the 
formula for L (488) becomes 

i = (2: — C)8in/± A: 

. .. sin J? . , , ^^ k sin 7r„ 

Hence, putting 

A' = A + (^-C)^4^ (630) 

we have 

i = (z - C) 5^^ zh ^ (631) 


When the angle i> is known, therefore, the preceding formulae 
will determine i, with which the computation will be carried 
out in precisely the same form as in the case of a solar eclipse, 
Art. 829. To find i?, let OP, Fig. 51, be drawn in the direction 
of the pole of the equinoctial ; then we have POQ = p, and, 
denoting POM by §, 

and Q has here the same signification as in the general equations 
(567), as shown in Art. 295 : so that when N and 4' have been 
found by (568) and (569), we have Q = N+'^, or 

^ = N+ 4— p (632) 

But to compute ^^ by (569) we must know i, and this involves 
-ff, which depends upon i?. The problem can, therefore, be 
solved only by successive approximations; but this is a very 
slight objection in the present case, since the only formulae to be 
repeated are those for L and ^'j and the second approximation 
will mostly be final. It can only be in a case such as the occul- 
tation of Saturn's ring, where the outline of the eclipsed body is 
very elliptical, and especially when the contact occurs near the 
northern or southern limb of the moon, that it may be necessary 
(for extreme accuracy) to compute H a second time and, conse- 
quently, '^ a third time. 

The formula (629) is adapted to the general case of an ellip- 


soidal body partially illuminated, the point of contact being on 
the defective limb. When the point of contact is on the full 
limb, we have only to put V= 0, and the formula becomes 

.. jT sin 3^8in i» 

8mg= (688) 

sm V 

and for the full limb of a spherical planet (Venus, Mercury, and 
Mars) we have H=^ s^. 

In the first approximation we may take L^= ± L 

354. Sometimes it may not be known from the record of the 
obser\'^ation whether the point of contact is on the full or the 
defective limb of the planet. This might be determined by the 
method of Art. 350 ; but, since that method supposes the position 
angle p' to be given, which we do not here employ, the following 
more direct and simple process may be used. In that article the 
common point of tangency of the two curves of the full and 
defective limbs was determined by the condition 

cos to ^ 

u. sm w + V. = 

in which u^ and i\ denotes the co-ordinates of the point of tan- 
gency. Li the notation of Art. 353 this is simply y^ = ; and 

since we have 

8 cos (»^, — w) cos' V 

^' ~ ^/[l — co8»(.?^ — w) sin« F] 
it follows that we must have 

cos (f?j — M7) = or t^j == M? qi 90° 

ITcnce, when, as in our present application, we take w = 90®, we 


\ = or ^^ = 180° 

Hence a point is to be regarded as on the east Umbfor values of d 
between 0° and 180°, and on tlie west limb for values of (? between 180° 
and 360° ; and (Art. 350) the east or the west limb is defective accord- 
ing as Vis between 0° and 180° or between 180° and 360°. 

But, since sin «?' and sin t? have the same sign, we deduce from 
this a still more simple rule ; for we have sin ;f = sin i?' sin F, 
whence it follows that the observed point is on the defective limb 
when sin ^ is positive^ and on the full limb when sin j[ is negative. 


855. In the cases of the planets Neptune, Uranus, and the 
asteroids, the oecultation of their centres will be observed, and 
it will be most convenient to compute by the method for a fixed 
star, only substituting for tt the difference of the moon's and 
planet's horizontal parallaxes — ^that is, the relative parallax — in 
the formulse for x and y, Art. 341. 

This artifice of using the relative parallax may also be used 
with advantage for Jupiter and Saturn. 

Having thus found x and ^ as for a fixed star, we shall have, 
in the preceding method, 

X = (2 — C) ^^ d: * (634) 

the other formulae remaining unchanged. 

Example 1. — Several occultations of Saturn's Ring were ob- 
served by Dr. E^ane at Van Rensselaer Harbor on the northwest 
coast of Greenland during the second Qrinnell Expedition in 
search of Sir John Franklin.* The first of these was as 
follows : 

1853 December 12th, Van Rensselaer Mean Time 
Immersion, contact of last point of ring, . . . 14* 20* 48'.8 
Emersion, « a u « ... 14 54 I8.3 

The assumed longitude of the place of observation was co = 4* 43* 32* 
west of Greenwich. The latitude was tp = 78° 37' 4", whence 

Jog p sin sp' = 9.989862 log p cos ^' = 9.296642 

I. From the Nautical Almanac we take for 1853 Dec. 12, 19*, 
;> = — 2° 37'.3 I = 24^ 0'.4 whence log c = log sin I = 9.6094 
and from page 578, the outer ring only being observed, 

«o = 187".66 log sin 5, = 6.9687 

* **A8tTonomical Obserrations in the Arctio Seas by Elibha Kent Kane, M.D., 
U.S.N. Reduced and discussed by Charles A. Sohott, Assistant U.S. Coast 
Surrey." Published by the Smithsonian Institution, May, 1860. 



n. We shall compute the elements of the occultation for the 
centre of the planet for the Greenwich hours 18*, 19*, and 20*. 
For these times we take the following quantities from the 
Nautical Almanac, applying to them the corrections determined 
by Mr. Schott from the Greenwich observations of this date : 


Gr. T. 





3» 36- 55'.23 

+ 18<' 2'47".5 

64' 7".68 


38 53.92 

12 13 .9 

7 .22 


40 52.81 

21 35 .7 

6 .76 







3» 39- 9'.88 

+ 17° 14' 28".4 





26 .5 



24 .5 

The corrections applied to the Nautical Almanac values to 
obtain the above are Aa = — 0'.22, a^ = — 5".0, Aa' = + O'.IS, 
a5' = — 8".9, a;: = + 0".3, this last correction being derived 
from Mr. Adams's Table in the Nautical Almanac for 1856. 

We shall use the relative parallax, and compute as for a fixed 
star, taking it — tz' for ;r, namely 



54' 6".73 


6 .17 


5 .71 

whence we find for the moon's co-ordinates. 

Gr. T. 








— 0.59152 


+ 0.45781 

+ 0.52457 
+ 0.52466 
+ 0.52475 

+ 0.89382 
+ 1.06817 
+ 1.24250 

+ 0.17436 
+ 0.17434 
+ 0.17432 



and, taking z = r = -: — for 19*, as sujficiently accurate. 

sin ic 

z = 63.54 

m. For the co-ordinates of the place of observation : 

Local mean time t 

t + w 

Local sid. time fi 
and hence, by the formate on p. 550, 



14» 20- 48'.8 

19 4 20.8 

IIT" 4' 59".7 

14* 54- 18'.3 

19 37 50.3 

125° 28' 44".7 



z — C 

+ 0.17529 
+ 0.90575 
+ 0.38 

+ 0.18685 
+ 0.91363 
-f 0.35 

IV. Assuming now two epochs corresponding nearly to the 
times of observation, the remainder of the computation in extenso 
is as follows : 





f = m BiuM 

71 =m cos M 



re' = n sin iV 

y' = n cos N 



Then, for a first appproximation, by the formula 



19*.07 — 

19*.63 — 

19* 4- 12* 

19* 37- 48'. 

— 0.03017 

+ 0.26365 

+ 1.08037 

+ 1.17800 

— 0.20546 

+ 0.07680 

+ 0.17462 

+ 0.26437 

310<» 21' 38" 

16^ 11' 56" 



+ 0.52467 

+ 0.52472 

+ 0.17434 

+ 0.17433 

71^ 37' 10" 

710 37' 20" 



sin 4 = 

m 8in(Jf — N) 

and observing that the immersion is here an interior contact and 
the emersion an exterior contact, we have 



(i = 

log sin (if — N) 

log m 

k) ar. CO. log L 

log sin 4 


log tan 1* 

log tan d' 

log sin 6 
ar. CO. log sin i^' 


flog a 

a^ k = 1/ 


sm 8. 




67^ 43'.2 
74 14.5 

131 67.7 






— 0.27264 

— 0.26708 



303<' 45'.5 
74 16.6 
18 0.1 






+ 0.27264 

+ 0.27612 


Applying the difference between log L and log k to log sin -i^, we 
find, for our second approximation, 

Corrected log sin 4 




59^ 39'.6 

304° 49'.5 

" i> 

133 54.1 

19 4.1 

log tan d 



log tan »»' 



log sin »> 



ar. CO. log sin ?>' 





Corrected log a 



" a 



" L 

— 0.26721 

+ 0.27618 

" log L 



Final value of log sin 4 



log cos 4. 



* The angle 4^ is to be taken so that L cos ^ shall be negative for immersion and 

positive for emersion, Art. 829. 

, ^ ... , ^. sin // sin 1^ 2 — ^ 
t Putting a = (z — C) =: ^ . *o 



h = 3600, log b = log 

A X cos 4 


log c = log 

hm cos ( Jf — IT) 




b — c = T 

Gr. Time of obs. = T^ + t= T 

If now we wish to form the equations of condition for deter- 
mining the effect of errors in the data, we proceed precisely as 
in the case of a solar eclipse, page 538, and find 








+ 1027'.3 


+ 1017 .2 

+ 31.0 

+ 10.1 

19* 4-'43-.0 

19* 37- 58M 

4 43 54.2 

4 43 39.8 



log y tan 4 
log V sec 4 



where log V = log 7-0 o 

depending on the correction of the parallax and of the eccen- 
tricity of the meridian, the equations of condition are 

(Im.) «i = 4* 43- 54-.2 — 2.001 y + 3.421 ^ — 3.965 r aA* 
(Em.) 01, = 4 43 39 .8 — 2.001 r — 2.881 d + 3.511 i:Ak 

Eliminating i> from these equations, we have 

w, = 4* 43- 46'.4 — 2.001 r + 0.092 r Ait 

An error of 1" in the moon's semidiameter (represented by Tt^k) 
would, therefore, have no sensible effect upon this combined 
result ; and since y must also be very small, as we have corrected 
the places of the moon and planet by the Greenwich observations, 
we can adopt, as the definite result from this observation, 

o>, = 4* 43- 46'.4 

It will be observed that in this example Oudemans's value, 
k = 0.27264, has been employed ; but our final equation shows 
that the result would have been sensibly the same if we had 
taken the usual value 0.27227 ; for the reduction of the result to 
that which the latter value of k would have given is only 
0.092 X 3247 X (— 0.00037) = — O-.ll. 



Example 2. — The occultation ofVenuSy April 24, 1860, was 
observed at the U. S. Military Academy, West Point (ai =4* SS"* 51', 
f = 41° 23' 31".2), and at Albany {w = 4* 54- 59'.4, f> = 42° 3^ 
49".5), as follows : 


First contact; planet's full limb 

Disappearance of cusp 

West Point. 
Sid. time. 

10* 46" 53-.35 
10 47 47.80 

Mean time. 

8»31- 1-.9 
8 31 54.2 

The observations were made with the large refiractors of the 
West Point and Dudley observatories. 

I. To find p for the cusp observations, we have for the Green- 
wich time 13*.478, which is the mean of the times of the obser- 
vations at the two places, and will serve for both. 

Planet, a' = 78° 38'.6 
Sun, A =32 45.5 

d' = 25° Sy.l 
D=13 12.9 

whence, by (623), 
and, from p. 578, 

— 7° 2r.3 


log sin 8^ = 5.6175 

n. We shall compute the moon's co-ordinates only for the 
Greenwich times 13*.4 and 13*.5. For these times the American 
Ephemeris furnishes the following data : 


Gr. T. 




13 .5 

79° 12' 16".8 
79 15 58 .5 

+ 26° 43' 1".6 
26 43 4 .3 

57' 6".6 
57 6 .7 

V «;uuo. 




13 5 

78° 38' 23".3 

78 38 40 .7 

+ 25° 59' 2".5 
25 59 4 .3